ABCF->ab-angle a

Percentage Accurate: 18.8% → 58.4%

Time: 31.3s

Alternatives: 36

Speedup: 3.0×

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: 58.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + C \cdot C\\ t_1 := \sqrt{\frac{1}{t\_0}}\\ t_2 := 1 - C \cdot t\_1\\ t_3 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_4 := C + \mathsf{hypot}\left(B\_m, C\right)\\ t_5 := \left(4 \cdot A\right) \cdot C\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_5\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_5 - {B\_m}^{2}}\\ t_7 := t\_5 - B\_m \cdot B\_m\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{-118}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + t\_3}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{elif}\;t\_6 \leq 20000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(B\_m \cdot B\_m\right)\right) \cdot \left(F \cdot t\_4\right) + A \cdot \left(2 \cdot \left(\left(A \cdot F\right) \cdot \left(\left(C \cdot -4\right) \cdot t\_2 + 0.5 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(t\_1 \cdot \left(1 - \frac{C \cdot C}{t\_0}\right)\right)\right)\right) + F \cdot \left(-4 \cdot \left(C \cdot t\_4\right) + \left(B\_m \cdot B\_m\right) \cdot t\_2\right)\right)\right)}}{t\_7}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + t\_3\right)\right)}^{0.5}}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \left({t\_4}^{-0.5} \cdot {F}^{-0.5}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* C C)))
        (t_1 (sqrt (/ 1.0 t_0)))
        (t_2 (- 1.0 (* C t_1)))
        (t_3 (hypot B_m (- A C)))
        (t_4 (+ C (hypot B_m C)))
        (t_5 (* (* 4.0 A) C))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_5) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_5 (pow B_m 2.0))))
        (t_7 (- t_5 (* B_m B_m))))
   (if (<= t_6 -5e-118)
     (-
      0.0
      (*
       (sqrt F)
       (pow (* 2.0 (/ (+ (+ A C) t_3) (+ (* B_m B_m) (* A (* C -4.0))))) 0.5)))
     (if (<= t_6 20000.0)
       (/
        (sqrt
         (+
          (* (* 2.0 (* B_m B_m)) (* F t_4))
          (*
           A
           (*
            2.0
            (+
             (*
              (* A F)
              (+
               (* (* C -4.0) t_2)
               (* 0.5 (* (* B_m B_m) (* t_1 (- 1.0 (/ (* C C) t_0)))))))
             (* F (+ (* -4.0 (* C t_4)) (* (* B_m B_m) t_2))))))))
        t_7)
       (if (<= t_6 INFINITY)
         (*
          (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
          (/ (pow (+ A (+ C t_3)) 0.5) t_7))
         (/ -1.0 (* (/ B_m (sqrt 2.0)) (* (pow t_4 -0.5) (pow F -0.5)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (C * C);
	double t_1 = sqrt((1.0 / t_0));
	double t_2 = 1.0 - (C * t_1);
	double t_3 = hypot(B_m, (A - C));
	double t_4 = C + hypot(B_m, C);
	double t_5 = (4.0 * A) * C;
	double t_6 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_5) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_5 - pow(B_m, 2.0));
	double t_7 = t_5 - (B_m * B_m);
	double tmp;
	if (t_6 <= -5e-118) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (((A + C) + t_3) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else if (t_6 <= 20000.0) {
		tmp = sqrt((((2.0 * (B_m * B_m)) * (F * t_4)) + (A * (2.0 * (((A * F) * (((C * -4.0) * t_2) + (0.5 * ((B_m * B_m) * (t_1 * (1.0 - ((C * C) / t_0))))))) + (F * ((-4.0 * (C * t_4)) + ((B_m * B_m) * t_2)))))))) / t_7;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + t_3)), 0.5) / t_7);
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * (pow(t_4, -0.5) * pow(F, -0.5)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (C * C);
	double t_1 = Math.sqrt((1.0 / t_0));
	double t_2 = 1.0 - (C * t_1);
	double t_3 = Math.hypot(B_m, (A - C));
	double t_4 = C + Math.hypot(B_m, C);
	double t_5 = (4.0 * A) * C;
	double t_6 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_5) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_5 - Math.pow(B_m, 2.0));
	double t_7 = t_5 - (B_m * B_m);
	double tmp;
	if (t_6 <= -5e-118) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (((A + C) + t_3) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else if (t_6 <= 20000.0) {
		tmp = Math.sqrt((((2.0 * (B_m * B_m)) * (F * t_4)) + (A * (2.0 * (((A * F) * (((C * -4.0) * t_2) + (0.5 * ((B_m * B_m) * (t_1 * (1.0 - ((C * C) / t_0))))))) + (F * ((-4.0 * (C * t_4)) + ((B_m * B_m) * t_2)))))))) / t_7;
	} else if (t_6 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + t_3)), 0.5) / t_7);
	} else {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * (Math.pow(t_4, -0.5) * Math.pow(F, -0.5)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) + (C * C)
	t_1 = math.sqrt((1.0 / t_0))
	t_2 = 1.0 - (C * t_1)
	t_3 = math.hypot(B_m, (A - C))
	t_4 = C + math.hypot(B_m, C)
	t_5 = (4.0 * A) * C
	t_6 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_5) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_5 - math.pow(B_m, 2.0))
	t_7 = t_5 - (B_m * B_m)
	tmp = 0
	if t_6 <= -5e-118:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (((A + C) + t_3) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	elif t_6 <= 20000.0:
		tmp = math.sqrt((((2.0 * (B_m * B_m)) * (F * t_4)) + (A * (2.0 * (((A * F) * (((C * -4.0) * t_2) + (0.5 * ((B_m * B_m) * (t_1 * (1.0 - ((C * C) / t_0))))))) + (F * ((-4.0 * (C * t_4)) + ((B_m * B_m) * t_2)))))))) / t_7
	elif t_6 <= math.inf:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + t_3)), 0.5) / t_7)
	else:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * (math.pow(t_4, -0.5) * math.pow(F, -0.5)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) + Float64(C * C))
	t_1 = sqrt(Float64(1.0 / t_0))
	t_2 = Float64(1.0 - Float64(C * t_1))
	t_3 = hypot(B_m, Float64(A - C))
	t_4 = Float64(C + hypot(B_m, C))
	t_5 = Float64(Float64(4.0 * A) * C)
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_5) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_5 - (B_m ^ 2.0)))
	t_7 = Float64(t_5 - Float64(B_m * B_m))
	tmp = 0.0
	if (t_6 <= -5e-118)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(Float64(Float64(A + C) + t_3) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	elseif (t_6 <= 20000.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * Float64(B_m * B_m)) * Float64(F * t_4)) + Float64(A * Float64(2.0 * Float64(Float64(Float64(A * F) * Float64(Float64(Float64(C * -4.0) * t_2) + Float64(0.5 * Float64(Float64(B_m * B_m) * Float64(t_1 * Float64(1.0 - Float64(Float64(C * C) / t_0))))))) + Float64(F * Float64(Float64(-4.0 * Float64(C * t_4)) + Float64(Float64(B_m * B_m) * t_2)))))))) / t_7);
	elseif (t_6 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + t_3)) ^ 0.5) / t_7));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * Float64((t_4 ^ -0.5) * (F ^ -0.5))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) + (C * C);
	t_1 = sqrt((1.0 / t_0));
	t_2 = 1.0 - (C * t_1);
	t_3 = hypot(B_m, (A - C));
	t_4 = C + hypot(B_m, C);
	t_5 = (4.0 * A) * C;
	t_6 = sqrt(((2.0 * (((B_m ^ 2.0) - t_5) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_5 - (B_m ^ 2.0));
	t_7 = t_5 - (B_m * B_m);
	tmp = 0.0;
	if (t_6 <= -5e-118)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (((A + C) + t_3) / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	elseif (t_6 <= 20000.0)
		tmp = sqrt((((2.0 * (B_m * B_m)) * (F * t_4)) + (A * (2.0 * (((A * F) * (((C * -4.0) * t_2) + (0.5 * ((B_m * B_m) * (t_1 * (1.0 - ((C * C) / t_0))))))) + (F * ((-4.0 * (C * t_4)) + ((B_m * B_m) * t_2)))))))) / t_7;
	elseif (t_6 <= Inf)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + t_3)) ^ 0.5) / t_7);
	else
		tmp = -1.0 / ((B_m / sqrt(2.0)) * ((t_4 ^ -0.5) * (F ^ -0.5)));
	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[(B$95$m * B$95$m), $MachinePrecision] + N[(C * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(C * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$5), $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$5 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e-118], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(N[(N[(A + C), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 20000.0], N[(N[Sqrt[N[(N[(N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(A * N[(2.0 * N[(N[(N[(A * F), $MachinePrecision] * N[(N[(N[(C * -4.0), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(N[(C * C), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * N[(N[(-4.0 * N[(C * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + t$95$3), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$4, -0.5], $MachinePrecision] * N[Power[F, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000015e-118

   1. Initial program 45.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 54.0%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 72.6%

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

   if -5.00000000000000015e-118 < (/.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))) < 2e4

   1. Initial program 30.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.5%

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

   5. Taylor expanded in A around 0

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

   7. Simplified 53.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot \left(B \cdot B\right)\right) \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right) + A \cdot \left(2 \cdot \left(\left(A \cdot F\right) \cdot \left(\left(-4 \cdot C\right) \cdot \left(1 + \left(-C\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right) + 0.5 \cdot \left(\left(B \cdot B\right) \cdot \left(\left(1 - \frac{C \cdot C}{B \cdot B + C \cdot C}\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)\right)\right) + F \cdot \left(-4 \cdot \left(C \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right) + \left(B \cdot B\right) \cdot \left(1 + \left(-C\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2e4 < (/.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 35.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 67.6%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 80.3%

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

   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 0.4%

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

   6. Applied egg-rr 0.7%

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

   7. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 2.0%

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

   9. Simplified 2.0%

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

       1. pow1/2 N/A

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

       2. inv-pow N/A

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

       3. pow-pow N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

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

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

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

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

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

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

       9. hypot-define N/A

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

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

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval 30.9%

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

   11. Applied egg-rr 30.9%

       \[\leadsto \frac{1}{-\frac{B}{\sqrt{2}} \cdot \color{blue}{\left({\left(C + \mathsf{hypot}\left(B, C\right)\right)}^{-0.5} \cdot {F}^{-0.5}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-118}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \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 20000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(B \cdot B\right)\right) \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right) + A \cdot \left(2 \cdot \left(\left(A \cdot F\right) \cdot \left(\left(C \cdot -4\right) \cdot \left(1 - C \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right) + 0.5 \cdot \left(\left(B \cdot B\right) \cdot \left(\sqrt{\frac{1}{B \cdot B + C \cdot C}} \cdot \left(1 - \frac{C \cdot C}{B \cdot B + C \cdot C}\right)\right)\right)\right) + F \cdot \left(-4 \cdot \left(C \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right) + \left(B \cdot B\right) \cdot \left(1 - C \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)\right)\right)\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:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \left({\left(C + \mathsf{hypot}\left(B, C\right)\right)}^{-0.5} \cdot {F}^{-0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-162}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + t\_0\right)\right)}^{0.5}}{t\_1 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \left({\left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}^{-0.5} \cdot {F}^{-0.5}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 5e-162)
     (-
      0.0
      (*
       (sqrt F)
       (pow (* 2.0 (/ (+ (+ A C) t_0) (+ (* B_m B_m) (* A (* C -4.0))))) 0.5)))
     (if (<= t_2 INFINITY)
       (*
        (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
        (/ (pow (+ A (+ C t_0)) 0.5) (- t_1 (* B_m B_m))))
       (/
        -1.0
        (*
         (/ B_m (sqrt 2.0))
         (* (pow (+ C (hypot B_m C)) -0.5) (pow F -0.5))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 5e-162) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + t_0)), 0.5) / (t_1 - (B_m * B_m)));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * (pow((C + hypot(B_m, C)), -0.5) * pow(F, -0.5)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.hypot(B_m, (A - C));
	double t_1 = (4.0 * A) * C;
	double t_2 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_1 - Math.pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 5e-162) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + t_0)), 0.5) / (t_1 - (B_m * B_m)));
	} else {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * (Math.pow((C + Math.hypot(B_m, C)), -0.5) * Math.pow(F, -0.5)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	t_1 = (4.0 * A) * C
	t_2 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_1 - math.pow(B_m, 2.0))
	tmp = 0
	if t_2 <= 5e-162:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	elif t_2 <= math.inf:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + t_0)), 0.5) / (t_1 - (B_m * B_m)))
	else:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * (math.pow((C + math.hypot(B_m, C)), -0.5) * math.pow(F, -0.5)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= 5e-162)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(Float64(Float64(A + C) + t_0) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + t_0)) ^ 0.5) / Float64(t_1 - Float64(B_m * B_m))));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * Float64((Float64(C + hypot(B_m, C)) ^ -0.5) * (F ^ -0.5))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = hypot(B_m, (A - C));
	t_1 = (4.0 * A) * C;
	t_2 = sqrt(((2.0 * (((B_m ^ 2.0) - t_1) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_1 - (B_m ^ 2.0));
	tmp = 0.0;
	if (t_2 <= 5e-162)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	elseif (t_2 <= Inf)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + t_0)) ^ 0.5) / (t_1 - (B_m * B_m)));
	else
		tmp = -1.0 / ((B_m / sqrt(2.0)) * (((C + hypot(B_m, C)) ^ -0.5) * (F ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-162], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(t$95$1 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[F, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000014e-162

   1. Initial program 38.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 44.4%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 60.9%

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

   if 5.00000000000000014e-162 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 52.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 75.0%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 81.0%

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

   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 0.4%

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

   6. Applied egg-rr 0.7%

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

   7. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{1}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 2.0%

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

   9. Simplified 2.0%

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

       1. pow1/2 N/A

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

       2. inv-pow N/A

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

       3. pow-pow N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

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

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

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

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

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

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

       9. hypot-define N/A

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

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

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval 30.9%

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

   11. Applied egg-rr 30.9%

       \[\leadsto \frac{1}{-\frac{B}{\sqrt{2}} \cdot \color{blue}{\left({\left(C + \mathsf{hypot}\left(B, C\right)\right)}^{-0.5} \cdot {F}^{-0.5}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{-162}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \left({\left(C + \mathsf{hypot}\left(B, C\right)\right)}^{-0.5} \cdot {F}^{-0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.75 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{B\_m}}{{2}^{-0.5}} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.75e+80)
   (*
    (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
    (/
     (pow (+ A (+ C (hypot B_m (- A C)))) 0.5)
     (- (* (* 4.0 A) C) (* B_m B_m))))
   (*
    (/ (/ -1.0 B_m) (pow 2.0 -0.5))
    (* (sqrt F) (sqrt (+ A (hypot B_m A)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.75e+80) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = ((-1.0 / B_m) / pow(2.0, -0.5)) * (sqrt(F) * sqrt((A + hypot(B_m, A))));
	}
	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 <= 2.75e+80) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + Math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = ((-1.0 / B_m) / Math.pow(2.0, -0.5)) * (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.75e+80:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)))
	else:
		tmp = ((-1.0 / B_m) / math.pow(2.0, -0.5)) * (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.75e+80)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) ^ 0.5) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) / (2.0 ^ -0.5)) * Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.75e+80)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + hypot(B_m, (A - C)))) ^ 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	else
		tmp = ((-1.0 / B_m) / (2.0 ^ -0.5)) * (sqrt(F) * sqrt((A + hypot(B_m, A))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.75e+80], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] / N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.74999999999999984e80

   1. Initial program 26.3%

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

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 34.0%

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

   if 2.74999999999999984e80 < B

   1. Initial program 5.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 47.3%

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

   5. Simplified 47.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval 70.5%

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

   9. Applied egg-rr 70.5%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.75 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{B}}{{2}^{-0.5}} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.3 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{-1}{B\_m \cdot \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.3e+79)
   (*
    (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
    (/
     (pow (+ A (+ C (hypot B_m (- A C)))) 0.5)
     (- (* (* 4.0 A) C) (* B_m B_m))))
   (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ -1.0 (* B_m (sqrt 0.5))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.3e+79) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (-1.0 / (B_m * sqrt(0.5)));
	}
	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 <= 5.3e+79) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + Math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (-1.0 / (B_m * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.3e+79:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)))
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (-1.0 / (B_m * math.sqrt(0.5)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.3e+79)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) ^ 0.5) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m * sqrt(0.5))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.3e+79)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + hypot(B_m, (A - C)))) ^ 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (-1.0 / (B_m * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.3e+79], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.29999999999999978e79

   1. Initial program 26.3%

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

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 34.0%

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

   if 5.29999999999999978e79 < B

   1. Initial program 5.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 47.3%

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

   5. Simplified 47.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval 70.5%

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

   9. Applied egg-rr 70.5%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{B \cdot \sqrt{\frac{1}{2}}}\right)}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   11. Step-by-step derivation

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

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

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

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

       3. sqrt-lowering-sqrt.f64 70.6%

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

   12. Simplified 70.6%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.3 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{-1}{B \cdot \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.5e+79)
   (*
    (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
    (/
     (pow (+ A (+ C (hypot B_m (- A C)))) 0.5)
     (- (* (* 4.0 A) C) (* B_m B_m))))
   (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ (sqrt 2.0) (- 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 <= 6.5e+79) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / (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 <= 6.5e+79) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + Math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / (0.0 - B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.5e+79:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + math.hypot(B_m, (A - C)))), 0.5) / (((4.0 * A) * C) - (B_m * B_m)))
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (math.sqrt(2.0) / (0.0 - B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.5e+79)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) ^ 0.5) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / 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 <= 6.5e+79)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + hypot(B_m, (A - C)))) ^ 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / (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, 6.5e+79], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.49999999999999954e79

   1. Initial program 26.3%

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

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 34.0%

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

   if 6.49999999999999954e79 < B

   1. Initial program 5.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 47.3%

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

   5. Simplified 47.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 70.5%

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

   9. Applied egg-rr 70.5%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 7 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + t\_0\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))))
   (if (<= B_m 7e+40)
     (*
      (sqrt (* 2.0 (* F (+ (* B_m B_m) (* C (* A -4.0))))))
      (/ (pow (+ A (+ C t_0)) 0.5) (- (* (* 4.0 A) C) (* B_m B_m))))
     (if (<= B_m 7.6e+124)
       (-
        0.0
        (*
         (sqrt F)
         (pow
          (* 2.0 (/ (+ (+ A C) t_0) (+ (* B_m B_m) (* A (* C -4.0)))))
          0.5)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 7e+40) {
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (pow((A + (C + t_0)), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 = Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 7e+40) {
		tmp = Math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (Math.pow((A + (C + t_0)), 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 7e+40:
		tmp = math.sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (math.pow((A + (C + t_0)), 0.5) / (((4.0 * A) * C) - (B_m * B_m)))
	elif B_m <= 7.6e+124:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (B_m <= 7e+40)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))))) * Float64((Float64(A + Float64(C + t_0)) ^ 0.5) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))));
	elseif (B_m <= 7.6e+124)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(Float64(Float64(A + C) + t_0) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 7e+40)
		tmp = sqrt((2.0 * (F * ((B_m * B_m) + (C * (A * -4.0)))))) * (((A + (C + t_0)) ^ 0.5) / (((4.0 * A) * C) - (B_m * B_m)));
	elseif (B_m <= 7.6e+124)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B$95$m, 7e+40], N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.6e+124], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.9999999999999998e40

   1. Initial program 26.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 30.6%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

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

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

   6. Applied egg-rr 34.3%

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

   if 6.9999999999999998e40 < B < 7.5999999999999997e124

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 39.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 48.9%

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

   if 7.5999999999999997e124 < B

   1. Initial program 2.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 49.1%

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

   5. Simplified 49.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 78.2%

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

   7. Applied egg-rr 78.2%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 0.00155:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + t\_0\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))))
   (if (<= B_m 0.00155)
     (/
      (*
       (sqrt (* F (+ (* B_m B_m) (* C (* A -4.0)))))
       (sqrt (* 2.0 (+ A (+ C t_0)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 7.6e+124)
       (-
        0.0
        (*
         (sqrt F)
         (pow
          (* 2.0 (/ (+ (+ A C) t_0) (+ (* B_m B_m) (* A (* C -4.0)))))
          0.5)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 0.00155) {
		tmp = (sqrt((F * ((B_m * B_m) + (C * (A * -4.0))))) * sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 = Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 0.00155) {
		tmp = (Math.sqrt((F * ((B_m * B_m) + (C * (A * -4.0))))) * Math.sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 0.00155:
		tmp = (math.sqrt((F * ((B_m * B_m) + (C * (A * -4.0))))) * math.sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 7.6e+124:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (B_m <= 0.00155)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))))) * sqrt(Float64(2.0 * Float64(A + Float64(C + t_0))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 7.6e+124)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(Float64(Float64(A + C) + t_0) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 0.00155)
		tmp = (sqrt((F * ((B_m * B_m) + (C * (A * -4.0))))) * sqrt((2.0 * (A + (C + t_0))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 7.6e+124)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B$95$m, 0.00155], N[(N[(N[Sqrt[N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $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[B$95$m, 7.6e+124], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.00154999999999999995

   1. Initial program 25.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.0%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

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

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

   6. Applied egg-rr 33.7%

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

   if 0.00154999999999999995 < B < 7.5999999999999997e124

   1. Initial program 29.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 42.1%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 49.4%

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

   if 7.5999999999999997e124 < B

   1. Initial program 2.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 49.1%

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

   5. Simplified 49.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 78.2%

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

   7. Applied egg-rr 78.2%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00155:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot B_m (- A C)))))
   (if (<= B_m 8.8e-161)
     (/
      (* (sqrt t_0) (sqrt (* A (* -8.0 (* C F)))))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 7.6e+124)
       (-
        0.0
        (*
         (sqrt F)
         (pow (* 2.0 (/ t_0 (+ (* B_m B_m) (* A (* C -4.0))))) 0.5)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (A + C) + hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 8.8e-161) {
		tmp = (sqrt(t_0) * sqrt((A * (-8.0 * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 = (A + C) + Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 8.8e-161) {
		tmp = (Math.sqrt(t_0) * Math.sqrt((A * (-8.0 * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (A + C) + math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 8.8e-161:
		tmp = (math.sqrt(t_0) * math.sqrt((A * (-8.0 * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 7.6e+124:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))
	tmp = 0.0
	if (B_m <= 8.8e-161)
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(A * Float64(-8.0 * Float64(C * F))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 7.6e+124)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(t_0 / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (A + C) + hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 8.8e-161)
		tmp = (sqrt(t_0) * sqrt((A * (-8.0 * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 7.6e+124)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.8e-161], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(A * N[(-8.0 * N[(C * F), $MachinePrecision]), $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[B$95$m, 7.6e+124], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$0 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.80000000000000009e-161

   1. Initial program 23.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.5%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left(C \cdot F\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(\left(A \cdot C\right) \cdot F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 19.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.5%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}} \cdot {\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)}^{\frac{1}{2}}\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A + C\right), \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \left(\mathsf{hypot}\left(B, A - C\right)\right)\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \left(A - C\right)\right)\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right), \left({\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right), \left(\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot C\right) \cdot F\right) \cdot -8\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 19.6%

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

   if 8.80000000000000009e-161 < B < 7.5999999999999997e124

   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. Add Preprocessing

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 38.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 48.1%

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

   if 7.5999999999999997e124 < B

   1. Initial program 2.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 49.1%

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

   5. Simplified 49.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 78.2%

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

   7. Applied egg-rr 78.2%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \sqrt{A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot B_m (- A C)))))
   (if (<= B_m 2.8e-221)
     (/ (sqrt (* t_0 (* -8.0 (* F (* A C))))) (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 7.6e+124)
       (-
        0.0
        (*
         (sqrt F)
         (pow (* 2.0 (/ t_0 (+ (* B_m B_m) (* A (* C -4.0))))) 0.5)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (A + C) + hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 2.8e-221) {
		tmp = sqrt((t_0 * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (sqrt(F) * pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 = (A + C) + Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 2.8e-221) {
		tmp = Math.sqrt((t_0 * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.6e+124) {
		tmp = 0.0 - (Math.sqrt(F) * Math.pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (A + C) + math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 2.8e-221:
		tmp = math.sqrt((t_0 * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 7.6e+124:
		tmp = 0.0 - (math.sqrt(F) * math.pow((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))), 0.5))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))
	tmp = 0.0
	if (B_m <= 2.8e-221)
		tmp = Float64(sqrt(Float64(t_0 * Float64(-8.0 * Float64(F * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 7.6e+124)
		tmp = Float64(0.0 - Float64(sqrt(F) * (Float64(2.0 * Float64(t_0 / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (A + C) + hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 2.8e-221)
		tmp = sqrt((t_0 * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 7.6e+124)
		tmp = 0.0 - (sqrt(F) * ((2.0 * (t_0 / ((B_m * B_m) + (A * (C * -4.0))))) ^ 0.5));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.8e-221], N[(N[Sqrt[N[(t$95$0 * N[(-8.0 * N[(F * N[(A * C), $MachinePrecision]), $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[B$95$m, 7.6e+124], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$0 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.80000000000000019e-221

   1. Initial program 24.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.5%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left(C \cdot F\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(\left(A \cdot C\right) \cdot F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 19.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.5%

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

   if 2.80000000000000019e-221 < B < 7.5999999999999997e124

   1. Initial program 29.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 31.0%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

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

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

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

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

   7. Applied egg-rr 42.5%

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

   if 7.5999999999999997e124 < B

   1. Initial program 2.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 49.1%

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

   5. Simplified 49.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 78.2%

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

   7. Applied egg-rr 78.2%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+124}:\\ \;\;\;\;0 - \sqrt{F} \cdot {\left(2 \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 3.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + t\_0\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 4 \cdot 10^{+100}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{\left(A + C\right) + t\_0}{B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))))
   (if (<= B_m 3.3e+36)
     (/
      (sqrt (* (* (+ (* B_m B_m) (* C (* A -4.0))) (+ A (+ C t_0))) (* 2.0 F)))
      (- (* (* 4.0 A) C) (* B_m B_m)))
     (if (<= B_m 4e+100)
       (-
        0.0
        (*
         (sqrt 2.0)
         (sqrt (* F (/ (+ (+ A C) t_0) (+ (* B_m B_m) (* A (* C -4.0))))))))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 3.3e+36) {
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + t_0))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4e+100) {
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0)))))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 = Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 3.3e+36) {
		tmp = Math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + t_0))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4e+100) {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((F * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0)))))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 3.3e+36:
		tmp = math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + t_0))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 4e+100:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((F * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0)))))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (B_m <= 3.3e+36)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(A + Float64(C + t_0))) * Float64(2.0 * F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 4e+100)
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_0) / Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 3.3e+36)
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + t_0))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 4e+100)
		tmp = 0.0 - (sqrt(2.0) * sqrt((F * (((A + C) + t_0) / ((B_m * B_m) + (A * (C * -4.0)))))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B$95$m, 3.3e+36], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $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, 4e+100], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] / 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[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.2999999999999999e36

   1. Initial program 25.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.3%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 30.1%

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

   if 3.2999999999999999e36 < B < 4.00000000000000006e100

   1. Initial program 29.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 45.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 52.1%

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

   if 4.00000000000000006e100 < B

   1. Initial program 4.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 49.3%

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

   5. Simplified 49.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 74.9%

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

   7. Applied egg-rr 74.9%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 66.5%

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

   10. Simplified 66.5%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+100}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{B \cdot B + A \cdot \left(C \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.8e+76)
   (/
    (sqrt
     (*
      (* (+ (* B_m B_m) (* C (* A -4.0))) (+ A (+ C (hypot B_m (- A C)))))
      (* 2.0 F)))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- 0.0 (sqrt B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.8e+76) {
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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 <= 3.8e+76) {
		tmp = Math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + Math.hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * (0.0 - Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.8e+76:
		tmp = math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + math.hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * (0.0 - math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.8e+76)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(2.0 * F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(0.0 - sqrt(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 <= 3.8e+76)
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * (0.0 - sqrt(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, 3.8e+76], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $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[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.80000000000000024e76

   1. Initial program 26.3%

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

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

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

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

   6. Applied egg-rr 30.4%

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

   if 3.80000000000000024e76 < B

   1. Initial program 5.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 47.3%

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

   5. Simplified 47.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{B}\right)}, \mathsf{sqrt.f64}\left(F\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 62.8%

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

   10. Simplified 62.8%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(0 - \sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(B\_m \cdot B\_m - t\_0\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= F 7e-294)
     (/
      (sqrt
       (* (+ (+ A C) (hypot B_m (- A C))) (* (* 2.0 F) (- (* B_m B_m) t_0))))
      (- t_0 (* B_m B_m)))
     (if (<= F 1.4e+46)
       (* (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (/ -1.0 B_m))
       (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (F <= 7e-294) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((2.0 * F) * ((B_m * B_m) - t_0)))) / (t_0 - (B_m * B_m));
	} else if (F <= 1.4e+46) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	}
	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 tmp;
	if (F <= 7e-294) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) * ((2.0 * F) * ((B_m * B_m) - t_0)))) / (t_0 - (B_m * B_m));
	} else if (F <= 1.4e+46) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if F <= 7e-294:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) * ((2.0 * F) * ((B_m * B_m) - t_0)))) / (t_0 - (B_m * B_m))
	elif F <= 1.4e+46:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m)
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (F <= 7e-294)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(Float64(2.0 * F) * Float64(Float64(B_m * B_m) - t_0)))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif (F <= 1.4e+46)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / sqrt(2.0))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (F <= 7e-294)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((2.0 * F) * ((B_m * B_m) - t_0)))) / (t_0 - (B_m * B_m));
	elseif (F <= 1.4e+46)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	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]}, If[LessEqual[F, 7e-294], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e+46], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 7.00000000000000064e-294

   1. Initial program 36.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 53.9%

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

   if 7.00000000000000064e-294 < F < 1.40000000000000009e46

   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. Add Preprocessing

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. hypot-define N/A

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

      13. hypot-lowering-hypot.f64 25.4%

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

   5. Simplified 25.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. hypot-define N/A

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

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

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 26.8%

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

   7. Applied egg-rr 26.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval 26.9%

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

   9. Applied egg-rr 26.9%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l* N/A

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

       2. mul-1-neg N/A

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

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

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

       4. div-inv N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow1/2 N/A

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

       8. associate-*l* N/A

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

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

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

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

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

       11. pow1/2 N/A

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

       12. sqrt-unprod N/A

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

       13. pow1/2 N/A

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

       14. pow-prod-down N/A

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

   11. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{-\frac{1}{B} \cdot {\left(2 \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)\right)}^{0.5}} \]

   if 1.40000000000000009e46 < F

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 19.2%

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

   6. Applied egg-rr 20.1%

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

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 21.2%

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

   9. Simplified 21.2%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 5.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 5.8e-296)
   (/
    (sqrt
     (*
      (* (+ (* B_m B_m) (* C (* A -4.0))) (+ A (+ C (hypot B_m (- A C)))))
      (* 2.0 F)))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 2e+46)
     (* (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (/ -1.0 B_m))
     (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 5.8e-296) {
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 2e+46) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 5.8e-296) {
		tmp = Math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + Math.hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 2e+46) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 5.8e-296:
		tmp = math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + math.hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 2e+46:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m)
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 5.8e-296)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(2.0 * F))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 2e+46)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / 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 <= 5.8e-296)
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (A + (C + hypot(B_m, (A - C))))) * (2.0 * F))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 2e+46)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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, 5.8e-296], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e+46], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 5.79999999999999965e-296

   1. Initial program 38.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 55.5%

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

      1. *-commutative N/A

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

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 55.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.79999999999999965e-296 < F < 2e46

   1. Initial program 20.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 25.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 25.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}}\right)}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 26.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   7. Applied egg-rr 26.6%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{\frac{B}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{B \cdot \frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\frac{1}{B}}{\frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{{2}^{\frac{1}{2}}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{-1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      11. metadata-eval 26.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   9. Applied egg-rr 26.7%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \frac{1}{{2}^{\frac{-1}{2}}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       5. pow-flip N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\frac{1}{2}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \sqrt{2}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{B} \cdot \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{B}\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       11. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       12. sqrt-unprod N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot {\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right)\right) \]

       14. pow-prod-down N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({\left(2 \cdot \left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]

   11. Applied egg-rr 25.3%

       \[\leadsto \color{blue}{-\frac{1}{B} \cdot {\left(2 \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)\right)}^{0.5}} \]

   if 2e46 < F

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 19.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{B}{F}}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{B}{F}\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{2}\right)\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 21.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Simplified 21.2%

      \[\leadsto \frac{1}{\color{blue}{-\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 7e-294)
   (/
    (sqrt (* (+ (+ A C) (hypot B_m (- A C))) (* -8.0 (* F (* A C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 3e+46)
     (* (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (/ -1.0 B_m))
     (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 7e-294) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 3e+46) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 7e-294) {
		tmp = Math.sqrt((((A + C) + Math.hypot(B_m, (A - C))) * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 3e+46) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 7e-294:
		tmp = math.sqrt((((A + C) + math.hypot(B_m, (A - C))) * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 3e+46:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m)
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 7e-294)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(-8.0 * Float64(F * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 3e+46)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / 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 <= 7e-294)
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * (-8.0 * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 3e+46)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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, 7e-294], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-8.0 * N[(F * N[(A * C), $MachinePrecision]), $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[F, 3e+46], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 7.00000000000000064e-294

   1. Initial program 36.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 53.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left(C \cdot F\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(\left(A \cdot C\right) \cdot F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 51.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)} \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.00000000000000064e-294 < F < 3.00000000000000023e46

   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. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 25.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 25.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}}\right)}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 26.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   7. Applied egg-rr 26.8%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{\frac{B}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{B \cdot \frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\frac{1}{B}}{\frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{{2}^{\frac{1}{2}}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{-1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      11. metadata-eval 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   9. Applied egg-rr 26.9%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \frac{1}{{2}^{\frac{-1}{2}}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       5. pow-flip N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\frac{1}{2}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \sqrt{2}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{B} \cdot \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{B}\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       11. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       12. sqrt-unprod N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot {\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right)\right) \]

       14. pow-prod-down N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({\left(2 \cdot \left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]

   11. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{-\frac{1}{B} \cdot {\left(2 \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)\right)}^{0.5}} \]

   if 3.00000000000000023e46 < F

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 19.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{B}{F}}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{B}{F}\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{2}\right)\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 21.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Simplified 21.2%

      \[\leadsto \frac{1}{\color{blue}{-\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 7e-294)
   (/
    (sqrt (* (* -8.0 (* A C)) (* F (+ (+ A C) (hypot B_m (- A C))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 4.2e+46)
     (* (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (/ -1.0 B_m))
     (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 7e-294) {
		tmp = sqrt(((-8.0 * (A * C)) * (F * ((A + C) + hypot(B_m, (A - C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 4.2e+46) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 7e-294) {
		tmp = Math.sqrt(((-8.0 * (A * C)) * (F * ((A + C) + Math.hypot(B_m, (A - C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 4.2e+46) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 7e-294:
		tmp = math.sqrt(((-8.0 * (A * C)) * (F * ((A + C) + math.hypot(B_m, (A - C)))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 4.2e+46:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m)
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 7e-294)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(A * C)) * Float64(F * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 4.2e+46)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / 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 <= 7e-294)
		tmp = sqrt(((-8.0 * (A * C)) * (F * ((A + C) + hypot(B_m, (A - C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 4.2e+46)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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, 7e-294], N[(N[Sqrt[N[(N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $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[F, 4.2e+46], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 7.00000000000000064e-294

   1. Initial program 36.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 53.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot \left(C \cdot F\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(\left(A \cdot C\right) \cdot F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\left(A \cdot C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 51.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)} \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) \cdot F + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(C + A\right) \cdot F + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-8 \cdot \left(A \cdot C\right)\right), \left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \left(A \cdot C\right)\right), \left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot \left(\left(C + A\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot \left(C + A\right) + F \cdot \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot \left(A + C\right) + F \cdot \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(F, \left(\left(A + C\right) + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 48.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.00000000000000064e-294 < F < 4.2e46

   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. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 25.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 25.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}}\right)}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 26.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   7. Applied egg-rr 26.8%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{\frac{B}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{B \cdot \frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\frac{1}{B}}{\frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{{2}^{\frac{1}{2}}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{-1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      11. metadata-eval 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   9. Applied egg-rr 26.9%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \frac{1}{{2}^{\frac{-1}{2}}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       5. pow-flip N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\frac{1}{2}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \sqrt{2}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{B} \cdot \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{B}\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       11. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       12. sqrt-unprod N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot {\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right)\right) \]

       14. pow-prod-down N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({\left(2 \cdot \left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]

   11. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{-\frac{1}{B} \cdot {\left(2 \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)\right)}^{0.5}} \]

   if 4.2e46 < F

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 19.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{B}{F}}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{B}{F}\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{2}\right)\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 21.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Simplified 21.2%

      \[\leadsto \frac{1}{\color{blue}{-\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.5e-291)
   (/ (sqrt (* (* C (* A C)) (* F -16.0))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 1.12e+46)
     (* (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (/ -1.0 B_m))
     (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.5e-291) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 1.12e+46) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.5e-291) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 1.12e+46) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m);
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.5e-291:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 1.12e+46:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) * (-1.0 / B_m)
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.5e-291)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 1.12e+46)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / 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 <= 2.5e-291)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 1.12e+46)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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, 2.5e-291], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.12e+46], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.5000000000000002e-291

   1. Initial program 35.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 52.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 21.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 21.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.5000000000000002e-291 < F < 1.12e46

   1. Initial program 20.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 25.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}}\right)}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   7. Applied egg-rr 26.9%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{\frac{B}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{B \cdot \frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\frac{1}{B}}{\frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{{2}^{\frac{1}{2}}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{-1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      11. metadata-eval 27.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   9. Applied egg-rr 27.0%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\frac{1}{B}}{{2}^{\frac{-1}{2}}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \frac{1}{{2}^{\frac{-1}{2}}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       5. pow-flip N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot {2}^{\frac{1}{2}}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       7. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\left(\frac{1}{B} \cdot \sqrt{2}\right) \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{B} \cdot \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{B}\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\sqrt{2} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       11. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right)\right) \]

       12. sqrt-unprod N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot \sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{1}{2}} \cdot {\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right)\right) \]

       14. pow-prod-down N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({\left(2 \cdot \left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]

   11. Applied egg-rr 25.6%

       \[\leadsto \color{blue}{-\frac{1}{B} \cdot {\left(2 \cdot \left(\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)\right)}^{0.5}} \]

   if 1.12e46 < F

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 19.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{B}{F}}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{B}{F}\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{2}\right)\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 21.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Simplified 21.2%

      \[\leadsto \frac{1}{\color{blue}{-\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{+46}:\\ \;\;\;\;{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B\_m} \cdot \sqrt{B\_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B\_m}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.5e-291)
   (/ (sqrt (* (* C (* A C)) (* F -16.0))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 5e-32)
     (* (/ (sqrt 2.0) (- 0.0 B_m)) (sqrt (* B_m F)))
     (/ 1.0 (* (sqrt (/ B_m F)) (/ -1.0 (sqrt 2.0)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.5e-291) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 5e-32) {
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m * F));
	} else {
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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 <= 2.5d-291) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (f <= 5d-32) then
        tmp = (sqrt(2.0d0) / (0.0d0 - b_m)) * sqrt((b_m * f))
    else
        tmp = 1.0d0 / (sqrt((b_m / f)) * ((-1.0d0) / 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 <= 2.5e-291) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 5e-32) {
		tmp = (Math.sqrt(2.0) / (0.0 - B_m)) * Math.sqrt((B_m * F));
	} else {
		tmp = 1.0 / (Math.sqrt((B_m / F)) * (-1.0 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.5e-291:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 5e-32:
		tmp = (math.sqrt(2.0) / (0.0 - B_m)) * math.sqrt((B_m * F))
	else:
		tmp = 1.0 / (math.sqrt((B_m / F)) * (-1.0 / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.5e-291)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 5e-32)
		tmp = Float64(Float64(sqrt(2.0) / Float64(0.0 - B_m)) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(B_m / F)) * Float64(-1.0 / 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 <= 2.5e-291)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 5e-32)
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m * F));
	else
		tmp = 1.0 / (sqrt((B_m / F)) * (-1.0 / 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, 2.5e-291], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-32], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.5000000000000002e-291

   1. Initial program 35.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 52.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 21.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 21.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.5000000000000002e-291 < F < 5e-32

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 27.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]

   6. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \color{blue}{\left(\sqrt{B \cdot F}\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(B \cdot F\right)\right)\right) \]

      2. *-lowering-*.f64 25.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right)\right) \]

   8. Simplified 25.0%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{B \cdot F}} \]

   if 5e-32 < F

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{B}{F}}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{B}{F}\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{2}\right)\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 19.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(B, F\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right)\right) \]

   9. Simplified 19.6%

      \[\leadsto \frac{1}{\color{blue}{-\sqrt{\frac{B}{F}} \cdot \frac{1}{\sqrt{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{B}{F}} \cdot \frac{-1}{\sqrt{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B\_m} \cdot \sqrt{B\_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{\frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.5e-291)
   (/ (sqrt (* (* C (* A C)) (* F -16.0))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= F 6.2e+67)
     (* (/ (sqrt 2.0) (- 0.0 B_m)) (sqrt (* B_m F)))
     (- 0.0 (* (sqrt 2.0) (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.5e-291) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 6.2e+67) {
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m * F));
	} else {
		tmp = 0.0 - (sqrt(2.0) * sqrt((F / 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 (f <= 2.5d-291) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (f <= 6.2d+67) then
        tmp = (sqrt(2.0d0) / (0.0d0 - b_m)) * sqrt((b_m * f))
    else
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt((f / 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 (F <= 2.5e-291) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (F <= 6.2e+67) {
		tmp = (Math.sqrt(2.0) / (0.0 - B_m)) * Math.sqrt((B_m * F));
	} else {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.5e-291:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	elif F <= 6.2e+67:
		tmp = (math.sqrt(2.0) / (0.0 - B_m)) * math.sqrt((B_m * F))
	else:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((F / B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.5e-291)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (F <= 6.2e+67)
		tmp = Float64(Float64(sqrt(2.0) / Float64(0.0 - B_m)) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.5e-291)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (F <= 6.2e+67)
		tmp = (sqrt(2.0) / (0.0 - B_m)) * sqrt((B_m * F));
	else
		tmp = 0.0 - (sqrt(2.0) * sqrt((F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.5e-291], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e+67], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.5000000000000002e-291

   1. Initial program 35.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 52.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 21.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 21.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 2.5000000000000002e-291 < F < 6.19999999999999992e67

   1. Initial program 22.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 25.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 25.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]

   6. Taylor expanded in A around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \color{blue}{\left(\sqrt{B \cdot F}\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(B \cdot F\right)\right)\right) \]

      2. *-lowering-*.f64 22.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(B, F\right)\right)\right) \]

   8. Simplified 22.1%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{B \cdot F}} \]

   if 6.19999999999999992e67 < 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. 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 21.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 21.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2}}{0 - B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 32.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + 2 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B\_m \cdot B\_m\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{\frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 1.05e-141)
     (/ (sqrt (* -16.0 (* C (* A (* C F))))) t_0)
     (if (<= B_m 6.2e-27)
       (/
        (sqrt
         (*
          F
          (+
           (* -16.0 (* C (* A A)))
           (*
            2.0
            (*
             (* B_m B_m)
             (+
              (* -2.0 (/ (* A C) (- A C)))
              (+
               (* 2.0 A)
               (*
                (* B_m B_m)
                (+
                 (* 0.5 (/ (* A C) (* (- A C) (* (- A C) (- A C)))))
                 (* 0.5 (/ 1.0 (- A C))))))))))))
        t_0)
       (- 0.0 (* (sqrt 2.0) (sqrt (/ F 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 <= 1.05e-141) {
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	} else if (B_m <= 6.2e-27) {
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * ((A - C) * (A - C))))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0;
	} else {
		tmp = 0.0 - (sqrt(2.0) * sqrt((F / 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 <= 1.05d-141) then
        tmp = sqrt(((-16.0d0) * (c * (a * (c * f))))) / t_0
    else if (b_m <= 6.2d-27) then
        tmp = sqrt((f * (((-16.0d0) * (c * (a * a))) + (2.0d0 * ((b_m * b_m) * (((-2.0d0) * ((a * c) / (a - c))) + ((2.0d0 * a) + ((b_m * b_m) * ((0.5d0 * ((a * c) / ((a - c) * ((a - c) * (a - c))))) + (0.5d0 * (1.0d0 / (a - c)))))))))))) / t_0
    else
        tmp = 0.0d0 - (sqrt(2.0d0) * sqrt((f / 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 <= 1.05e-141) {
		tmp = Math.sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	} else if (B_m <= 6.2e-27) {
		tmp = Math.sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * ((A - C) * (A - C))))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0;
	} else {
		tmp = 0.0 - (Math.sqrt(2.0) * Math.sqrt((F / 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 <= 1.05e-141:
		tmp = math.sqrt((-16.0 * (C * (A * (C * F))))) / t_0
	elif B_m <= 6.2e-27:
		tmp = math.sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * ((A - C) * (A - C))))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0
	else:
		tmp = 0.0 - (math.sqrt(2.0) * math.sqrt((F / 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 <= 1.05e-141)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(A * Float64(C * F))))) / t_0);
	elseif (B_m <= 6.2e-27)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(-16.0 * Float64(C * Float64(A * A))) + Float64(2.0 * Float64(Float64(B_m * B_m) * Float64(Float64(-2.0 * Float64(Float64(A * C) / Float64(A - C))) + Float64(Float64(2.0 * A) + Float64(Float64(B_m * B_m) * Float64(Float64(0.5 * Float64(Float64(A * C) / Float64(Float64(A - C) * Float64(Float64(A - C) * Float64(A - C))))) + Float64(0.5 * Float64(1.0 / Float64(A - C)))))))))))) / t_0);
	else
		tmp = Float64(0.0 - Float64(sqrt(2.0) * sqrt(Float64(F / 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 <= 1.05e-141)
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	elseif (B_m <= 6.2e-27)
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * ((A - C) * (A - C))))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0;
	else
		tmp = 0.0 - (sqrt(2.0) * sqrt((F / 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, 1.05e-141], N[(N[Sqrt[N[(-16.0 * N[(C * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 6.2e-27], N[(N[Sqrt[N[(F * N[(N[(-16.0 * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(-2.0 * N[(N[(A * C), $MachinePrecision] / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * A), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(A * C), $MachinePrecision] / N[(N[(A - C), $MachinePrecision] * N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.05e-141

   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 28.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 12.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.9%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 16.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 16.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(C \cdot \left(\left(F \cdot C\right) \cdot A\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.05e-141 < B < 6.1999999999999997e-27

   1. Initial program 32.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 32.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 23.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sqrt{F \cdot \left(-16 \cdot \left({A}^{2} \cdot C\right) + 2 \cdot \left({B}^{2} \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + {B}^{2} \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 27.2%

       \[\leadsto \frac{\color{blue}{\sqrt{F \cdot \left(-16 \cdot \left(\left(A \cdot A\right) \cdot C\right) + 2 \cdot \left(\left(B \cdot B\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B \cdot B\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 6.1999999999999997e-27 < B

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      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 37.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 37.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + 2 \cdot \left(\left(B \cdot B\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B \cdot B\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;A \leq -6.5 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B\_m}} \cdot \frac{-1}{\sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -6.5e-82)
   (* (sqrt (/ (* F -0.5) A)) (- 0.0 (sqrt 2.0)))
   (* (sqrt (/ F B_m)) (/ -1.0 (sqrt 0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -6.5e-82) {
		tmp = sqrt(((F * -0.5) / A)) * (0.0 - sqrt(2.0));
	} else {
		tmp = sqrt((F / B_m)) * (-1.0 / sqrt(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 (a <= (-6.5d-82)) then
        tmp = sqrt(((f * (-0.5d0)) / a)) * (0.0d0 - sqrt(2.0d0))
    else
        tmp = sqrt((f / b_m)) * ((-1.0d0) / sqrt(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 (A <= -6.5e-82) {
		tmp = Math.sqrt(((F * -0.5) / A)) * (0.0 - Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt((F / B_m)) * (-1.0 / Math.sqrt(0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if A <= -6.5e-82:
		tmp = math.sqrt(((F * -0.5) / A)) * (0.0 - math.sqrt(2.0))
	else:
		tmp = math.sqrt((F / B_m)) * (-1.0 / math.sqrt(0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -6.5e-82)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(0.0 - sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(F / B_m)) * Float64(-1.0 / sqrt(0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -6.5e-82)
		tmp = sqrt(((F * -0.5) / A)) * (0.0 - sqrt(2.0));
	else
		tmp = sqrt((F / B_m)) * (-1.0 / sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -6.5e-82], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -6.4999999999999997e-82

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \left(\sqrt{2}\right)\right)\right) \]

   5. Simplified 10.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}} \]

   6. Taylor expanded in C around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{F}{A}\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{2} \cdot F}{A}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot F\right), A\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. *-lowering-*.f64 36.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, F\right), A\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   8. Simplified 36.4%

      \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{A}}} \cdot \sqrt{2} \]

   if -6.4999999999999997e-82 < A

   1. Initial program 28.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 22.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right)\right) \]

   5. Simplified 22.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F\right)}^{\frac{1}{2}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left({\left(A + \sqrt{B \cdot B + A \cdot A}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({F}^{\frac{1}{2}}\right)}\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({\color{blue}{F}}^{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 29.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   7. Applied egg-rr 29.8%

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{\frac{B}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{1}{B \cdot \frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\frac{1}{B}}{\frac{1}{\sqrt{2}}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{B}\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{\sqrt{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left(\frac{1}{{2}^{\frac{1}{2}}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\frac{-1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \left({2}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

      11. metadata-eval 29.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, B\right), \mathsf{pow.f64}\left(2, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   9. Applied egg-rr 29.8%

      \[\leadsto \left(-1 \cdot \color{blue}{\frac{\frac{1}{B}}{{2}^{-0.5}}}\right) \cdot \left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \]

   10. Taylor expanded in B around inf

       \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \frac{1}{\sqrt{\frac{1}{2}}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \frac{1}{\sqrt{\frac{1}{2}}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \frac{1}{\sqrt{\frac{1}{2}}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\frac{1}{\sqrt{\frac{1}{2}}}\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\frac{1}{\sqrt{\frac{1}{2}}}\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\frac{1}{\sqrt{\frac{1}{2}}}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{\frac{1}{2}}\right)\right)\right)\right) \]

       7. sqrt-lowering-sqrt.f64 17.6%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right) \]

   12. Simplified 17.6%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \frac{1}{\sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6.5 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \frac{-1}{\sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 18.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ t_1 := \left(A - C\right) \cdot \left(A - C\right)\\ \mathbf{if}\;A \leq -5.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -6.8 \cdot 10^{-140}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B\_m} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot t\_1\right)}{B\_m \cdot B\_m}\right)\right)}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + 2 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B\_m \cdot B\_m\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot t\_1} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (* (- A C) (- A C))))
   (if (<= A -5.5e+135)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -6.8e-140)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 3.7e-170)
         (/
          (sqrt
           (*
            (* B_m (* B_m B_m))
            (+
             (* 2.0 F)
             (*
              2.0
              (+
               (/ (* F (+ A C)) B_m)
               (/ (* F (+ (* -4.0 (* A C)) (* 0.5 t_1))) (* B_m B_m)))))))
          (- 0.0 (* B_m B_m)))
         (/
          (sqrt
           (*
            F
            (+
             (* -16.0 (* C (* A A)))
             (*
              2.0
              (*
               (* B_m B_m)
               (+
                (* -2.0 (/ (* A C) (- A C)))
                (+
                 (* 2.0 A)
                 (*
                  (* B_m B_m)
                  (+
                   (* 0.5 (/ (* A C) (* (- A C) t_1)))
                   (* 0.5 (/ 1.0 (- A C))))))))))))
          t_0))))))

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 = (A - C) * (A - C);
	double tmp;
	if (A <= -5.5e+135) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -6.8e-140) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 3.7e-170) {
		tmp = sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * t_1))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * t_1))) + (0.5 * (1.0 / (A - C)))))))))))) / t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    t_1 = (a - c) * (a - c)
    if (a <= (-5.5d+135)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-6.8d-140)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 3.7d-170) then
        tmp = sqrt(((b_m * (b_m * b_m)) * ((2.0d0 * f) + (2.0d0 * (((f * (a + c)) / b_m) + ((f * (((-4.0d0) * (a * c)) + (0.5d0 * t_1))) / (b_m * b_m))))))) / (0.0d0 - (b_m * b_m))
    else
        tmp = sqrt((f * (((-16.0d0) * (c * (a * a))) + (2.0d0 * ((b_m * b_m) * (((-2.0d0) * ((a * c) / (a - c))) + ((2.0d0 * a) + ((b_m * b_m) * ((0.5d0 * ((a * c) / ((a - c) * t_1))) + (0.5d0 * (1.0d0 / (a - c)))))))))))) / t_0
    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 = (A - C) * (A - C);
	double tmp;
	if (A <= -5.5e+135) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -6.8e-140) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 3.7e-170) {
		tmp = Math.sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * t_1))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * t_1))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0;
	}
	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 = (A - C) * (A - C)
	tmp = 0
	if A <= -5.5e+135:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -6.8e-140:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 3.7e-170:
		tmp = math.sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * t_1))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m))
	else:
		tmp = math.sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * t_1))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0
	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))
	t_1 = Float64(Float64(A - C) * Float64(A - C))
	tmp = 0.0
	if (A <= -5.5e+135)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -6.8e-140)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 3.7e-170)
		tmp = Float64(sqrt(Float64(Float64(B_m * Float64(B_m * B_m)) * Float64(Float64(2.0 * F) + Float64(2.0 * Float64(Float64(Float64(F * Float64(A + C)) / B_m) + Float64(Float64(F * Float64(Float64(-4.0 * Float64(A * C)) + Float64(0.5 * t_1))) / Float64(B_m * B_m))))))) / Float64(0.0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(-16.0 * Float64(C * Float64(A * A))) + Float64(2.0 * Float64(Float64(B_m * B_m) * Float64(Float64(-2.0 * Float64(Float64(A * C) / Float64(A - C))) + Float64(Float64(2.0 * A) + Float64(Float64(B_m * B_m) * Float64(Float64(0.5 * Float64(Float64(A * C) / Float64(Float64(A - C) * t_1))) + Float64(0.5 * Float64(1.0 / Float64(A - C)))))))))))) / t_0);
	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 = (A - C) * (A - C);
	tmp = 0.0;
	if (A <= -5.5e+135)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -6.8e-140)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 3.7e-170)
		tmp = sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * t_1))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	else
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (2.0 * ((B_m * B_m) * ((-2.0 * ((A * C) / (A - C))) + ((2.0 * A) + ((B_m * B_m) * ((0.5 * ((A * C) / ((A - C) * t_1))) + (0.5 * (1.0 / (A - C)))))))))))) / t_0;
	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]}, Block[{t$95$1 = N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -5.5e+135], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -6.8e-140], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 3.7e-170], N[(N[Sqrt[N[(N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] + N[(2.0 * N[(N[(N[(F * N[(A + C), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(N[(F * N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(-16.0 * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(-2.0 * N[(N[(A * C), $MachinePrecision] / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * A), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(A * C), $MachinePrecision] / N[(N[(A - C), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -5.4999999999999999e135

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -5.4999999999999999e135 < A < -6.80000000000000017e-140

   1. Initial program 13.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 9.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 9.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 9.8%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.4%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -6.80000000000000017e-140 < A < 3.7e-170

   1. Initial program 32.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 36.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \color{blue}{\left(-1 \cdot {B}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left({B}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \left(B \cdot \color{blue}{B}\right)\right)\right) \]

      3. *-lowering-*.f64 29.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right)\right) \]

   7. Simplified 29.5%

      \[\leadsto \frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{-1 \cdot \left(B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{F \cdot \left(A + C\right)}{B}\right), \left(\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 15.0%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)\right)}}}{-1 \cdot \left(B \cdot B\right)} \]

   if 3.7e-170 < A

   1. Initial program 27.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 32.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 20.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sqrt{F \cdot \left(-16 \cdot \left({A}^{2} \cdot C\right) + 2 \cdot \left({B}^{2} \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + {B}^{2} \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 24.0%

       \[\leadsto \frac{\color{blue}{\sqrt{F \cdot \left(-16 \cdot \left(\left(A \cdot A\right) \cdot C\right) + 2 \cdot \left(\left(B \cdot B\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B \cdot B\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -6.8 \cdot 10^{-140}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + 2 \cdot \left(\left(B \cdot B\right) \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + \left(2 \cdot A + \left(B \cdot B\right) \cdot \left(0.5 \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)} + 0.5 \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 18.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -6.5 \cdot 10^{-140}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B\_m} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B\_m \cdot B\_m}\right)\right)}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(2 \cdot \left(B\_m \cdot B\_m\right) + -4 \cdot \left(C \cdot \left(C \cdot 0\right)\right)\right)\right)}{A}\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -4e+135)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -6.5e-140)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 1.7e-59)
         (/
          (sqrt
           (*
            (* B_m (* B_m B_m))
            (+
             (* 2.0 F)
             (*
              2.0
              (+
               (/ (* F (+ A C)) B_m)
               (/
                (* F (+ (* -4.0 (* A C)) (* 0.5 (* (- A C) (- A C)))))
                (* B_m B_m)))))))
          (- 0.0 (* B_m B_m)))
         (/
          (sqrt
           (*
            (* A A)
            (+
             (* -16.0 (* C F))
             (/
              (* 2.0 (* F (+ (* 2.0 (* B_m B_m)) (* -4.0 (* C (* C 0.0))))))
              A))))
          t_0))))))

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 (A <= -4e+135) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -6.5e-140) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.7e-59) {
		tmp = sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * ((A - C) * (A - C))))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-4d+135)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-6.5d-140)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 1.7d-59) then
        tmp = sqrt(((b_m * (b_m * b_m)) * ((2.0d0 * f) + (2.0d0 * (((f * (a + c)) / b_m) + ((f * (((-4.0d0) * (a * c)) + (0.5d0 * ((a - c) * (a - c))))) / (b_m * b_m))))))) / (0.0d0 - (b_m * b_m))
    else
        tmp = sqrt(((a * a) * (((-16.0d0) * (c * f)) + ((2.0d0 * (f * ((2.0d0 * (b_m * b_m)) + ((-4.0d0) * (c * (c * 0.0d0)))))) / a)))) / t_0
    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 (A <= -4e+135) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -6.5e-140) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.7e-59) {
		tmp = Math.sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * ((A - C) * (A - C))))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0;
	}
	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 A <= -4e+135:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -6.5e-140:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 1.7e-59:
		tmp = math.sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * ((A - C) * (A - C))))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m))
	else:
		tmp = math.sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0
	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 (A <= -4e+135)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -6.5e-140)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 1.7e-59)
		tmp = Float64(sqrt(Float64(Float64(B_m * Float64(B_m * B_m)) * Float64(Float64(2.0 * F) + Float64(2.0 * Float64(Float64(Float64(F * Float64(A + C)) / B_m) + Float64(Float64(F * Float64(Float64(-4.0 * Float64(A * C)) + Float64(0.5 * Float64(Float64(A - C) * Float64(A - C))))) / Float64(B_m * B_m))))))) / Float64(0.0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(Float64(A * A) * Float64(Float64(-16.0 * Float64(C * F)) + Float64(Float64(2.0 * Float64(F * Float64(Float64(2.0 * Float64(B_m * B_m)) + Float64(-4.0 * Float64(C * Float64(C * 0.0)))))) / A)))) / t_0);
	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 (A <= -4e+135)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -6.5e-140)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 1.7e-59)
		tmp = sqrt(((B_m * (B_m * B_m)) * ((2.0 * F) + (2.0 * (((F * (A + C)) / B_m) + ((F * ((-4.0 * (A * C)) + (0.5 * ((A - C) * (A - C))))) / (B_m * B_m))))))) / (0.0 - (B_m * B_m));
	else
		tmp = sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0;
	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[A, -4e+135], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -6.5e-140], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.7e-59], N[(N[Sqrt[N[(N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] + N[(2.0 * N[(N[(N[(F * N[(A + C), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(N[(F * N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(A * A), $MachinePrecision] * N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(F * N[(N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(C * N[(C * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -3.99999999999999985e135

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -3.99999999999999985e135 < A < -6.4999999999999995e-140

   1. Initial program 13.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 9.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 9.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 9.8%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.4%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -6.4999999999999995e-140 < A < 1.70000000000000009e-59

   1. Initial program 30.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 33.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \color{blue}{\left(-1 \cdot {B}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left({B}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \left(B \cdot \color{blue}{B}\right)\right)\right) \]

      3. *-lowering-*.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right)\right) \]

   7. Simplified 26.8%

      \[\leadsto \frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{-1 \cdot \left(B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({B}^{3} \cdot \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)}\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{3}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \left(2 \cdot F + \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\left(2 \cdot F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(2 \cdot \frac{F \cdot \left(A + C\right)}{B} + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \left(2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, F\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{F \cdot \left(A + C\right)}{B}\right), \left(\frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + \frac{1}{2} \cdot {\left(A - C\right)}^{2}\right)}{{B}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 12.1%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)\right)}}}{-1 \cdot \left(B \cdot B\right)} \]

   if 1.70000000000000009e-59 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(\frac{2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 24.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)\right)}{A}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -6.5 \cdot 10^{-140}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot \left(B \cdot B\right)\right) \cdot \left(2 \cdot F + 2 \cdot \left(\frac{F \cdot \left(A + C\right)}{B} + \frac{F \cdot \left(-4 \cdot \left(A \cdot C\right) + 0.5 \cdot \left(\left(A - C\right) \cdot \left(A - C\right)\right)\right)}{B \cdot B}\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(2 \cdot \left(B \cdot B\right) + -4 \cdot \left(C \cdot \left(C \cdot 0\right)\right)\right)\right)}{A}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 18.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -3 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.65 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(2 \cdot \left(B\_m \cdot B\_m\right) + -4 \cdot \left(C \cdot \left(C \cdot 0\right)\right)\right)\right)}{A}\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -3e+136)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -1.9e-136)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 1.65e-60)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (/
          (sqrt
           (*
            (* A A)
            (+
             (* -16.0 (* C F))
             (/
              (* 2.0 (* F (+ (* 2.0 (* B_m B_m)) (* -4.0 (* C (* C 0.0))))))
              A))))
          t_0))))))

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 (A <= -3e+136) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1.9e-136) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.65e-60) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-3d+136)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-1.9d-136)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 1.65d-60) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((a * a) * (((-16.0d0) * (c * f)) + ((2.0d0 * (f * ((2.0d0 * (b_m * b_m)) + ((-4.0d0) * (c * (c * 0.0d0)))))) / a)))) / t_0
    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 (A <= -3e+136) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1.9e-136) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.65e-60) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0;
	}
	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 A <= -3e+136:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -1.9e-136:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 1.65e-60:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0
	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 (A <= -3e+136)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -1.9e-136)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 1.65e-60)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(A * A) * Float64(Float64(-16.0 * Float64(C * F)) + Float64(Float64(2.0 * Float64(F * Float64(Float64(2.0 * Float64(B_m * B_m)) + Float64(-4.0 * Float64(C * Float64(C * 0.0)))))) / A)))) / t_0);
	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 (A <= -3e+136)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -1.9e-136)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 1.65e-60)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt(((A * A) * ((-16.0 * (C * F)) + ((2.0 * (F * ((2.0 * (B_m * B_m)) + (-4.0 * (C * (C * 0.0)))))) / A)))) / t_0;
	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[A, -3e+136], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -1.9e-136], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.65e-60], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(A * A), $MachinePrecision] * N[(N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(F * N[(N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(C * N[(C * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -2.99999999999999979e136

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -2.99999999999999979e136 < A < -1.9000000000000001e-136

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -1.9000000000000001e-136 < A < 1.6499999999999999e-60

   1. Initial program 30.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.6499999999999999e-60 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(\frac{2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)\right)}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 24.7%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)\right)}{A}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.65 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + \frac{2 \cdot \left(F \cdot \left(2 \cdot \left(B \cdot B\right) + -4 \cdot \left(C \cdot \left(C \cdot 0\right)\right)\right)\right)}{A}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 18.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -4.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -1 \cdot 10^{-134}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)} \cdot \sqrt{F \cdot \left(C \cdot \left(-16 \cdot \left(A \cdot A\right)\right)\right) + B\_m \cdot \left(B\_m \cdot \left(\frac{F \cdot \left(B\_m \cdot B\_m\right)}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}\\ \end{array} \end{array} \]

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))))
   (if (<= A -4.3e+135)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -1e-134)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 9.2e-61)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (*
          (/ -1.0 (+ (* B_m B_m) (* C (* A -4.0))))
          (sqrt
           (+
            (* F (* C (* -16.0 (* A A))))
            (* B_m (* B_m (+ (/ (* F (* B_m B_m)) A) (* 4.0 (* A F)))))))))))))

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 (A <= -4.3e+135) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1e-134) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 9.2e-61) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = (-1.0 / ((B_m * B_m) + (C * (A * -4.0)))) * sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * 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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-4.3d+135)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-1d-134)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 9.2d-61) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = ((-1.0d0) / ((b_m * b_m) + (c * (a * (-4.0d0))))) * sqrt(((f * (c * ((-16.0d0) * (a * a)))) + (b_m * (b_m * (((f * (b_m * b_m)) / a) + (4.0d0 * (a * 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 t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -4.3e+135) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1e-134) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 9.2e-61) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = (-1.0 / ((B_m * B_m) + (C * (A * -4.0)))) * Math.sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F)))))));
	}
	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 A <= -4.3e+135:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -1e-134:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 9.2e-61:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = (-1.0 / ((B_m * B_m) + (C * (A * -4.0)))) * math.sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F)))))))
	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 (A <= -4.3e+135)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -1e-134)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 9.2e-61)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0)))) * sqrt(Float64(Float64(F * Float64(C * Float64(-16.0 * Float64(A * A)))) + Float64(B_m * Float64(B_m * Float64(Float64(Float64(F * Float64(B_m * B_m)) / A) + Float64(4.0 * Float64(A * F))))))));
	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 (A <= -4.3e+135)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -1e-134)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 9.2e-61)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = (-1.0 / ((B_m * B_m) + (C * (A * -4.0)))) * sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F)))))));
	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[A, -4.3e+135], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -1e-134], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 9.2e-61], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(F * N[(C * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B$95$m * N[(B$95$m * N[(N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -4.29999999999999972e135

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -4.29999999999999972e135 < A < -1.00000000000000004e-134

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -1.00000000000000004e-134 < A < 9.19999999999999967e-61

   1. Initial program 30.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.19999999999999967e-61 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 22.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 22.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 22.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   12. Applied egg-rr 22.5%

       \[\leadsto \color{blue}{\frac{-1}{B \cdot B + C \cdot \left(A \cdot -4\right)} \cdot \sqrt{F \cdot \left(C \cdot \left(\left(A \cdot A\right) \cdot -16\right)\right) + B \cdot \left(B \cdot \left(\frac{\left(B \cdot B\right) \cdot F}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -1 \cdot 10^{-134}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot B + C \cdot \left(A \cdot -4\right)} \cdot \sqrt{F \cdot \left(C \cdot \left(-16 \cdot \left(A \cdot A\right)\right)\right) + B \cdot \left(B \cdot \left(\frac{F \cdot \left(B \cdot B\right)}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 18.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -2.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -1 \cdot 10^{-134}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot \left(-16 \cdot \left(A \cdot A\right)\right)\right) + B\_m \cdot \left(B\_m \cdot \left(\frac{F \cdot \left(B\_m \cdot B\_m\right)}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -2.1e+136)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -1e-134)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 4.6e-60)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (/
          (sqrt
           (+
            (* F (* C (* -16.0 (* A A))))
            (* B_m (* B_m (+ (/ (* F (* B_m B_m)) A) (* 4.0 (* A F)))))))
          t_0))))))

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 (A <= -2.1e+136) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1e-134) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.6e-60) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F))))))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-2.1d+136)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-1d-134)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 4.6d-60) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((f * (c * ((-16.0d0) * (a * a)))) + (b_m * (b_m * (((f * (b_m * b_m)) / a) + (4.0d0 * (a * f))))))) / t_0
    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 (A <= -2.1e+136) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1e-134) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.6e-60) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F))))))) / t_0;
	}
	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 A <= -2.1e+136:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -1e-134:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 4.6e-60:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F))))))) / t_0
	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 (A <= -2.1e+136)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -1e-134)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 4.6e-60)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(F * Float64(C * Float64(-16.0 * Float64(A * A)))) + Float64(B_m * Float64(B_m * Float64(Float64(Float64(F * Float64(B_m * B_m)) / A) + Float64(4.0 * Float64(A * F))))))) / t_0);
	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 (A <= -2.1e+136)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -1e-134)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 4.6e-60)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt(((F * (C * (-16.0 * (A * A)))) + (B_m * (B_m * (((F * (B_m * B_m)) / A) + (4.0 * (A * F))))))) / t_0;
	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[A, -2.1e+136], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -1e-134], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.6e-60], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(F * N[(C * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B$95$m * N[(B$95$m * N[(N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -2.0999999999999999e136

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -2.0999999999999999e136 < A < -1.00000000000000004e-134

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -1.00000000000000004e-134 < A < 4.6000000000000003e-60

   1. Initial program 30.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.6000000000000003e-60 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 22.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 22.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 22.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   11. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(-16 \cdot \left(A \cdot A\right)\right) \cdot C\right) \cdot F\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(F \cdot \left(\left(-16 \cdot \left(A \cdot A\right)\right) \cdot C\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \left(\left(-16 \cdot \left(A \cdot A\right)\right) \cdot C\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot \left(-16 \cdot \left(A \cdot A\right)\right)\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \left(-16 \cdot \left(A \cdot A\right)\right)\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \left(\left(A \cdot A\right) \cdot -16\right)\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(A \cdot A\right), -16\right)\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), -16\right)\right)\right), \left(\left(B \cdot B\right) \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), -16\right)\right)\right), \left(B \cdot \left(B \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), -16\right)\right)\right), \mathsf{*.f64}\left(B, \left(B \cdot \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), -16\right)\right)\right), \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       14. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), -16\right)\right)\right), \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, \left(2 \cdot \left(\frac{1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A}\right) + 2 \cdot \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   12. Applied egg-rr 22.4%

       \[\leadsto \frac{\color{blue}{\sqrt{F \cdot \left(C \cdot \left(\left(A \cdot A\right) \cdot -16\right)\right) + B \cdot \left(B \cdot \left(\frac{\left(B \cdot B\right) \cdot F}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -1 \cdot 10^{-134}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot \left(-16 \cdot \left(A \cdot A\right)\right)\right) + B \cdot \left(B \cdot \left(\frac{F \cdot \left(B \cdot B\right)}{A} + 4 \cdot \left(A \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 18.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -1.15 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + \frac{\left(B\_m \cdot B\_m\right) \cdot \left(B\_m \cdot B\_m + 4 \cdot \left(A \cdot A\right)\right)}{A}\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= A -1.15e+136)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -4.6e-133)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 4.5e-59)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (*
          (sqrt
           (*
            F
            (+
             (* -16.0 (* C (* A A)))
             (/ (* (* B_m B_m) (+ (* B_m B_m) (* 4.0 (* A A)))) A))))
          (/ 1.0 (- (* 4.0 (* A C)) (* 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 (A <= -1.15e+136) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -4.6e-133) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.5e-59) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (((B_m * B_m) * ((B_m * B_m) + (4.0 * (A * A)))) / A)))) * (1.0 / ((4.0 * (A * 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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-1.15d+136)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-4.6d-133)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 4.5d-59) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt((f * (((-16.0d0) * (c * (a * a))) + (((b_m * b_m) * ((b_m * b_m) + (4.0d0 * (a * a)))) / a)))) * (1.0d0 / ((4.0d0 * (a * 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 t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -1.15e+136) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -4.6e-133) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.5e-59) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt((F * ((-16.0 * (C * (A * A))) + (((B_m * B_m) * ((B_m * B_m) + (4.0 * (A * A)))) / A)))) * (1.0 / ((4.0 * (A * C)) - (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 A <= -1.15e+136:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -4.6e-133:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 4.5e-59:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt((F * ((-16.0 * (C * (A * A))) + (((B_m * B_m) * ((B_m * B_m) + (4.0 * (A * A)))) / A)))) * (1.0 / ((4.0 * (A * C)) - (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 (A <= -1.15e+136)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -4.6e-133)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 4.5e-59)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(-16.0 * Float64(C * Float64(A * A))) + Float64(Float64(Float64(B_m * B_m) * Float64(Float64(B_m * B_m) + Float64(4.0 * Float64(A * A)))) / A)))) * Float64(1.0 / Float64(Float64(4.0 * Float64(A * C)) - Float64(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 (A <= -1.15e+136)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -4.6e-133)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 4.5e-59)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt((F * ((-16.0 * (C * (A * A))) + (((B_m * B_m) * ((B_m * B_m) + (4.0 * (A * A)))) / A)))) * (1.0 / ((4.0 * (A * C)) - (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[A, -1.15e+136], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -4.6e-133], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.5e-59], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(-16.0 * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(4.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $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]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -1.15e136

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -1.15e136 < A < -4.6000000000000001e-133

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -4.6000000000000001e-133 < A < 4.50000000000000012e-59

   1. Initial program 30.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.50000000000000012e-59 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 22.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 22.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 22.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \color{blue}{\left(\frac{4 \cdot \left({A}^{2} \cdot F\right) + {B}^{2} \cdot F}{A}\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\left(4 \cdot \left({A}^{2} \cdot F\right) + {B}^{2} \cdot F\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\left({B}^{2} \cdot F + 4 \cdot \left({A}^{2} \cdot F\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({B}^{2} \cdot F\right), \left(4 \cdot \left({A}^{2} \cdot F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), \left(4 \cdot \left({A}^{2} \cdot F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), \left(4 \cdot \left({A}^{2} \cdot F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left(4 \cdot \left({A}^{2} \cdot F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(4, \left({A}^{2} \cdot F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\left({A}^{2}\right), F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\left(A \cdot A\right), F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 21.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), F\right)\right)\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   13. Simplified 21.1%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \color{blue}{\frac{\left(B \cdot B\right) \cdot F + 4 \cdot \left(\left(A \cdot A\right) \cdot F\right)}{A}}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   14. Taylor expanded in F around 0

       \[\leadsto \color{blue}{\sqrt{F \cdot \left(-16 \cdot \left({A}^{2} \cdot C\right) + \frac{{B}^{2} \cdot \left(4 \cdot {A}^{2} + {B}^{2}\right)}{A}\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(-16 \cdot \left({A}^{2} \cdot C\right) + \frac{{B}^{2} \cdot \left(4 \cdot {A}^{2} + {B}^{2}\right)}{A}\right)}\right), \color{blue}{\left(\frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\right)}\right) \]

   16. Simplified 22.4%

       \[\leadsto \color{blue}{\sqrt{F \cdot \left(-16 \cdot \left(\left(A \cdot A\right) \cdot C\right) + \frac{\left(B \cdot B\right) \cdot \left(4 \cdot \left(A \cdot A\right) + B \cdot B\right)}{A}\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.15 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right) + \frac{\left(B \cdot B\right) \cdot \left(B \cdot B + 4 \cdot \left(A \cdot A\right)\right)}{A}\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 18.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -1.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -4.8 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right) + \left(B\_m \cdot B\_m\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -1.6e+136)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -4.8e-133)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 1.85e-60)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (/
          (sqrt
           (+ (* (* C F) (* -16.0 (* A A))) (* (* B_m B_m) (* 4.0 (* A F)))))
          t_0))))))

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 (A <= -1.6e+136) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -4.8e-133) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.85e-60) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt((((C * F) * (-16.0 * (A * A))) + ((B_m * B_m) * (4.0 * (A * F))))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-1.6d+136)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-4.8d-133)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 1.85d-60) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt((((c * f) * ((-16.0d0) * (a * a))) + ((b_m * b_m) * (4.0d0 * (a * f))))) / t_0
    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 (A <= -1.6e+136) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -4.8e-133) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 1.85e-60) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt((((C * F) * (-16.0 * (A * A))) + ((B_m * B_m) * (4.0 * (A * F))))) / t_0;
	}
	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 A <= -1.6e+136:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -4.8e-133:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 1.85e-60:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt((((C * F) * (-16.0 * (A * A))) + ((B_m * B_m) * (4.0 * (A * F))))) / t_0
	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 (A <= -1.6e+136)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -4.8e-133)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 1.85e-60)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(C * F) * Float64(-16.0 * Float64(A * A))) + Float64(Float64(B_m * B_m) * Float64(4.0 * Float64(A * F))))) / t_0);
	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 (A <= -1.6e+136)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -4.8e-133)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 1.85e-60)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt((((C * F) * (-16.0 * (A * A))) + ((B_m * B_m) * (4.0 * (A * F))))) / t_0;
	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[A, -1.6e+136], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -4.8e-133], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.85e-60], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(C * F), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -1.59999999999999994e136

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -1.59999999999999994e136 < A < -4.8e-133

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -4.8e-133 < A < 1.85000000000000012e-60

   1. Initial program 30.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.85000000000000012e-60 < A

   1. Initial program 28.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 22.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 22.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 22.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. *-lowering-*.f64 22.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   13. Simplified 22.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -4.8 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right) + \left(B \cdot B\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 17.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -9.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-138}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}\\ \end{array} \end{array} \]

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))))
   (if (<= A -9.5e+135)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -1.3e-138)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 3.6e-35)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (/ 1.0 (/ t_0 (sqrt (* -16.0 (* (* A A) (* C F)))))))))))

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 (A <= -9.5e+135) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1.3e-138) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 3.6e-35) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 1.0 / (t_0 / sqrt((-16.0 * ((A * A) * (C * 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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-9.5d+135)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-1.3d-138)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 3.6d-35) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = 1.0d0 / (t_0 / sqrt(((-16.0d0) * ((a * a) * (c * 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 t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (A <= -9.5e+135) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -1.3e-138) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 3.6e-35) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 1.0 / (t_0 / Math.sqrt((-16.0 * ((A * A) * (C * F)))));
	}
	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 A <= -9.5e+135:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -1.3e-138:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 3.6e-35:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = 1.0 / (t_0 / math.sqrt((-16.0 * ((A * A) * (C * F)))))
	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 (A <= -9.5e+135)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -1.3e-138)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 3.6e-35)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(1.0 / Float64(t_0 / sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F))))));
	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 (A <= -9.5e+135)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -1.3e-138)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 3.6e-35)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = 1.0 / (t_0 / sqrt((-16.0 * ((A * A) * (C * F)))));
	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[A, -9.5e+135], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -1.3e-138], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 3.6e-35], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(t$95$0 / N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -9.50000000000000036e135

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -9.50000000000000036e135 < A < -1.3e-138

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -1.3e-138 < A < 3.60000000000000019e-35

   1. Initial program 32.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.0%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 3.60000000000000019e-35 < A

   1. Initial program 25.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 29.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2}\right), \left(C \cdot F\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot A\right), \left(C \cdot F\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(C \cdot F\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(C, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 20.0%

      \[\leadsto \frac{1}{\frac{C \cdot \left(A \cdot 4\right) - B \cdot B}{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-138}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(4 \cdot A\right) \cdot C - B \cdot B}{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 17.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -2.4 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t\_0}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -2.4e+136)
     (/ (sqrt (* (* C (* A C)) (* F -16.0))) t_0)
     (if (<= A -2.8e-133)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
       (if (<= A 4.2e-35)
         (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
         (/ (sqrt (* (* C F) (* -16.0 (* A A)))) t_0))))))

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 (A <= -2.4e+136) {
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -2.8e-133) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.2e-35) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-2.4d+136)) then
        tmp = sqrt(((c * (a * c)) * (f * (-16.0d0)))) / t_0
    else if (a <= (-2.8d-133)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 4.2d-35) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((c * f) * ((-16.0d0) * (a * a)))) / t_0
    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 (A <= -2.4e+136) {
		tmp = Math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	} else if (A <= -2.8e-133) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.2e-35) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt(((C * F) * (-16.0 * (A * A)))) / t_0;
	}
	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 A <= -2.4e+136:
		tmp = math.sqrt(((C * (A * C)) * (F * -16.0))) / t_0
	elif A <= -2.8e-133:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 4.2e-35:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt(((C * F) * (-16.0 * (A * A)))) / t_0
	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 (A <= -2.4e+136)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))) / t_0);
	elseif (A <= -2.8e-133)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 4.2e-35)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(C * F) * Float64(-16.0 * Float64(A * A)))) / t_0);
	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 (A <= -2.4e+136)
		tmp = sqrt(((C * (A * C)) * (F * -16.0))) / t_0;
	elseif (A <= -2.8e-133)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 4.2e-35)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / t_0;
	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[A, -2.4e+136], N[(N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[A, -2.8e-133], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.2e-35], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -2.4e136

   1. Initial program 1.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 2.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 18.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 36.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), C\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 36.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot C\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -2.4e136 < A < -2.7999999999999999e-133

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 10.0%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 0.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 0.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 28.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 28.9%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -2.7999999999999999e-133 < A < 4.2e-35

   1. Initial program 32.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.0%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.2e-35 < A

   1. Initial program 25.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot {A}^{2}\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2}\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 19.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.4 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 17.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;A \leq -5.5 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

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 (<= A -5.5e-133)
     (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
     (if (<= A 4.7e-35)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (/ (sqrt (* (* C F) (* -16.0 (* A A)))) t_0)))))

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 (A <= -5.5e-133) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.7e-35) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (a <= (-5.5d-133)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else if (a <= 4.7d-35) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = sqrt(((c * f) * ((-16.0d0) * (a * a)))) / t_0
    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 (A <= -5.5e-133) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else if (A <= 4.7e-35) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = Math.sqrt(((C * F) * (-16.0 * (A * A)))) / t_0;
	}
	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 A <= -5.5e-133:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	elif A <= 4.7e-35:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = math.sqrt(((C * F) * (-16.0 * (A * A)))) / t_0
	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 (A <= -5.5e-133)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	elseif (A <= 4.7e-35)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(C * F) * Float64(-16.0 * Float64(A * A)))) / t_0);
	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 (A <= -5.5e-133)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	elseif (A <= 4.7e-35)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / t_0;
	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[A, -5.5e-133], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.7e-35], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if A < -5.49999999999999977e-133

   1. Initial program 8.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 9.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 6.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 6.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 6.1%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 1.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 1.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 27.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 27.4%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -5.49999999999999977e-133 < A < 4.7e-35

   1. Initial program 32.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.0%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 4.7e-35 < A

   1. Initial program 25.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-16 \cdot {A}^{2}\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2}\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \left(C \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 19.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{elif}\;A \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 17.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 7.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 9.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \end{array} \end{array} \]

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 7.4e-71)
     (/ (sqrt (* -16.0 (* C (* A (* C F))))) t_0)
     (if (<= B_m 9.8e+78)
       (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) t_0)
       (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))))))

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 <= 7.4e-71) {
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	} else if (B_m <= 9.8e+78) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	}
	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 <= 7.4d-71) then
        tmp = sqrt(((-16.0d0) * (c * (a * (c * f))))) / t_0
    else if (b_m <= 9.8d+78) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / t_0
    else
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    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 <= 7.4e-71) {
		tmp = Math.sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	} else if (B_m <= 9.8e+78) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	}
	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 <= 7.4e-71:
		tmp = math.sqrt((-16.0 * (C * (A * (C * F))))) / t_0
	elif B_m <= 9.8e+78:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0
	else:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	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 <= 7.4e-71)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(A * Float64(C * F))))) / t_0);
	elseif (B_m <= 9.8e+78)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	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 <= 7.4e-71)
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / t_0;
	elseif (B_m <= 9.8e+78)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	else
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	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, 7.4e-71], N[(N[Sqrt[N[(-16.0 * N[(C * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 9.8e+78], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.3999999999999993e-71

   1. Initial program 24.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 12.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.1%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 16.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 16.8%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(C \cdot \left(\left(F \cdot C\right) \cdot A\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.3999999999999993e-71 < B < 9.8000000000000004e78

   1. Initial program 36.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 36.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 27.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 27.1%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.8000000000000004e78 < B

   1. Initial program 5.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 5.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 0.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 0.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 3.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 3.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 9.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 9.8%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 17.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.45e-69)
   (/ (sqrt (* -16.0 (* C (* A (* C F))))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 4e+77)
     (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) (- 0.0 (* B_m B_m)))
     (- 0.0 (pow (* (/ F A) (/ F A)) 0.25)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.45e-69) {
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4e+77) {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	}
	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 <= 1.45d-69) then
        tmp = sqrt(((-16.0d0) * (c * (a * (c * f))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (b_m <= 4d+77) then
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / (0.0d0 - (b_m * b_m))
    else
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    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 <= 1.45e-69) {
		tmp = Math.sqrt((-16.0 * (C * (A * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 4e+77) {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m));
	} else {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.45e-69:
		tmp = math.sqrt((-16.0 * (C * (A * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 4e+77:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m))
	else:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.45e-69)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(A * Float64(C * F))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 4e+77)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / Float64(0.0 - Float64(B_m * B_m)));
	else
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.45e-69)
		tmp = sqrt((-16.0 * (C * (A * (C * F))))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 4e+77)
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m));
	else
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.45e-69], N[(N[Sqrt[N[(-16.0 * N[(C * N[(A * N[(C * F), $MachinePrecision]), $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[B$95$m, 4e+77], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.4499999999999999e-69

   1. Initial program 24.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left({C}^{2}\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\left(C \cdot C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 12.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, C\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.1%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot A\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \left(\left(C \cdot F\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(C \cdot F\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\left(F \cdot C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 16.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, C\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 16.8%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(C \cdot \left(\left(F \cdot C\right) \cdot A\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.4499999999999999e-69 < B < 3.99999999999999993e77

   1. Initial program 36.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 36.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \color{blue}{\left(-1 \cdot {B}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left({B}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \left(B \cdot \color{blue}{B}\right)\right)\right) \]

      3. *-lowering-*.f64 33.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right)\right) \]

   7. Simplified 33.6%

      \[\leadsto \frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{-1 \cdot \left(B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 26.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 26.6%

       \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{-1 \cdot \left(B \cdot B\right)} \]

   if 3.99999999999999993e77 < B

   1. Initial program 5.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 5.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 0.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 0.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 3.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 3.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 9.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 9.8%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{0 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 13.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;A \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{0 - B\_m \cdot B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -4.2e-133)
   (- 0.0 (pow (* (/ F A) (/ F A)) 0.25))
   (/ (sqrt (* 2.0 (* F (* B_m (* B_m B_m))))) (- 0.0 (* B_m B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4.2e-133) {
		tmp = 0.0 - pow(((F / A) * (F / A)), 0.25);
	} else {
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (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 (a <= (-4.2d-133)) then
        tmp = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
    else
        tmp = sqrt((2.0d0 * (f * (b_m * (b_m * b_m))))) / (0.0d0 - (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 (A <= -4.2e-133) {
		tmp = 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
	} else {
		tmp = Math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if A <= -4.2e-133:
		tmp = 0.0 - math.pow(((F / A) * (F / A)), 0.25)
	else:
		tmp = math.sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (B_m * B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -4.2e-133)
		tmp = Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / Float64(0.0 - 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 (A <= -4.2e-133)
		tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
	else
		tmp = sqrt((2.0 * (F * (B_m * (B_m * B_m))))) / (0.0 - (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[A, -4.2e-133], N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.2000000000000002e-133

   1. Initial program 8.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 9.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 6.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 6.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 6.1%

       \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   11. Taylor expanded in A around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

       4. /-lowering-/.f64 1.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

   13. Simplified 1.1%

       \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
   14. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

       8. metadata-eval 27.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

   15. Applied egg-rr 27.4%

       \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

   if -4.2000000000000002e-133 < A

   1. Initial program 29.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \color{blue}{\left(-1 \cdot {B}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left({B}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \left(B \cdot \color{blue}{B}\right)\right)\right) \]

      3. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{B}\right)\right)\right) \]

   7. Simplified 18.3%

      \[\leadsto \frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{-1 \cdot \left(B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{3} \cdot F\right)\right)}\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({B}^{3} \cdot F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({B}^{3}\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(B \cdot {B}^{2}\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left({B}^{2}\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \left(B \cdot B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 5.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, \mathsf{*.f64}\left(B, B\right)\right), F\right)\right)\right), \mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   10. Simplified 5.9%

       \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{-1 \cdot \left(B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{0 - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 11.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (pow (* (/ F A) (/ F A)) 0.25)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - pow(((F / A) * (F / A)), 0.25);
}

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 = 0.0d0 - (((f / a) * (f / a)) ** 0.25d0)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.pow(((F / A) * (F / A)), 0.25);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.pow(((F / A) * (F / A)), 0.25)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - (Float64(Float64(F / A) * Float64(F / A)) ^ 0.25))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - (((F / A) * (F / A)) ^ 0.25);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Power[N[(N[(F / A), $MachinePrecision] * N[(F / A), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

5. Taylor expanded in B around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 10.7%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Taylor expanded in C around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. *-lowering-*.f64 10.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

10. Simplified 10.2%

    \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

11. Taylor expanded in A around 0

    \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

    2. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

    3. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

    4. /-lowering-/.f64 2.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

13. Simplified 2.9%

    \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
14. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

    2. sqr-pow N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

    3. pow-prod-down N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

    4. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{F}{A} \cdot \frac{F}{A}\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{A}\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \left(\frac{F}{A}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

    8. metadata-eval 12.5%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, A\right), \mathsf{/.f64}\left(F, A\right)\right), \frac{1}{4}\right)\right) \]

15. Applied egg-rr 12.5%

    \[\leadsto -\color{blue}{{\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25}} \]

16. Final simplification 12.5%

    \[\leadsto 0 - {\left(\frac{F}{A} \cdot \frac{F}{A}\right)}^{0.25} \]
17. Add Preprocessing

Alternative 35: 2.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - {\left(\frac{A}{F}\right)}^{-0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (pow (/ A F) -0.5)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - pow((A / F), -0.5);
}

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 = 0.0d0 - ((a / f) ** (-0.5d0))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.pow((A / F), -0.5);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.pow((A / F), -0.5)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - (Float64(A / F) ^ -0.5))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - ((A / F) ^ -0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Power[N[(A / F), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

5. Taylor expanded in B around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 10.7%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Taylor expanded in C around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. *-lowering-*.f64 10.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

10. Simplified 10.2%

    \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

11. Taylor expanded in A around 0

    \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

    2. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

    3. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

    4. /-lowering-/.f64 2.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

13. Simplified 2.9%

    \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]
14. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{F}{A}\right)}^{\frac{1}{2}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{\frac{A}{F}}\right)}^{\frac{1}{2}}\right)\right) \]

    3. inv-pow N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left({\left(\frac{A}{F}\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right) \]

    4. pow-pow N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{A}{F}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{A}{F}\right)}^{\frac{-1}{2}}\right)\right) \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{A}{F}\right), \frac{-1}{2}\right)\right) \]

    7. /-lowering-/.f64 3.0%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(A, F\right), \frac{-1}{2}\right)\right) \]

15. Applied egg-rr 3.0%

    \[\leadsto -\color{blue}{{\left(\frac{A}{F}\right)}^{-0.5}} \]

16. Final simplification 3.0%

    \[\leadsto 0 - {\left(\frac{A}{F}\right)}^{-0.5} \]
17. Add Preprocessing

Alternative 36: 2.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - \sqrt{\frac{F}{A}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (sqrt (/ F A))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt((F / A));
}

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 = 0.0d0 - sqrt((f / a))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.sqrt((F / A));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.sqrt((F / A))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(Float64(F / A)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - sqrt((F / A));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[(F / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

5. Taylor expanded in B around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + {B}^{2} \cdot \left(2 \cdot \left(F \cdot \left(-2 \cdot \frac{A \cdot C}{A - C} + 2 \cdot A\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\frac{1}{2} \cdot \frac{A \cdot C}{{\left(A - C\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{A - C}\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 10.7%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A + \frac{-2 \cdot \left(A \cdot C\right)}{A - C}\right) + \left(\left(B \cdot B\right) \cdot F\right) \cdot \left(\frac{0.5}{A - C} + \frac{0.5}{A - C} \cdot \frac{A \cdot C}{\left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Taylor expanded in C around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{{B}^{2} \cdot F}{A}\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{B}^{2} \cdot F}{A}\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. *-lowering-*.f64 10.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(A, A\right)\right), \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), A\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

10. Simplified 10.2%

    \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right) + \left(B \cdot B\right) \cdot \left(2 \cdot \color{blue}{\left(0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A} + 2 \cdot \left(A \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

11. Taylor expanded in A around 0

    \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{A}}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{A}}\right) \]

    2. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{A}}\right)\right) \]

    3. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{A}\right)\right)\right) \]

    4. /-lowering-/.f64 2.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, A\right)\right)\right) \]

13. Simplified 2.9%

    \[\leadsto \color{blue}{-\sqrt{\frac{F}{A}}} \]

14. Final simplification 2.9%

    \[\leadsto 0 - \sqrt{\frac{F}{A}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))