Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 81.2%

Time: 29.4s

Alternatives: 26

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{M\_m \cdot D\_m}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{{\left(0 - \ell\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(h \cdot \frac{\frac{-0.5}{t\_0}}{\ell}\right)}{d \cdot 4}\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-216}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot t\_0} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (* M_m D_m))))
   (if (<= d -7.8e-230)
     (*
      (/ (pow (- 0.0 d) 0.5) (pow (- 0.0 l) 0.5))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* M_m D_m) (* h (/ (/ -0.5 t_0) l))) (* d 4.0)))))
     (if (<= d 1.05e-216)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 6.8e-79)
         (*
          (+ 1.0 (/ (* (/ -0.5 (* l t_0)) (* h (* M_m D_m))) (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 7.5e+169)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           (/
            (*
             (sqrt d)
             (*
              (sqrt (/ d l))
              (-
               1.0
               (/
                (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
                l))))
            (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double tmp;
	if (d <= -7.8e-230) {
		tmp = (pow((0.0 - d), 0.5) / pow((0.0 - l), 0.5)) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0))));
	} else if (d <= 1.05e-216) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 6.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 7.5e+169) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (sqrt(d) * (sqrt((d / l)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / (m_m * d_m)
    if (d <= (-7.8d-230)) then
        tmp = (((0.0d0 - d) ** 0.5d0) / ((0.0d0 - l) ** 0.5d0)) * (sqrt((d / h)) * (1.0d0 + (((m_m * d_m) * (h * (((-0.5d0) / t_0) / l))) / (d * 4.0d0))))
    else if (d <= 1.05d-216) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 6.8d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * t_0)) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 7.5d+169) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (sqrt(d) * (sqrt((d / l)) * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)))) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double tmp;
	if (d <= -7.8e-230) {
		tmp = (Math.pow((0.0 - d), 0.5) / Math.pow((0.0 - l), 0.5)) * (Math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0))));
	} else if (d <= 1.05e-216) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 6.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 7.5e+169) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(d) * (Math.sqrt((d / l)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d / (M_m * D_m)
	tmp = 0
	if d <= -7.8e-230:
		tmp = (math.pow((0.0 - d), 0.5) / math.pow((0.0 - l), 0.5)) * (math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0))))
	elif d <= 1.05e-216:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 6.8e-79:
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 7.5e+169:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (math.sqrt(d) * (math.sqrt((d / l)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / Float64(M_m * D_m))
	tmp = 0.0
	if (d <= -7.8e-230)
		tmp = Float64(Float64((Float64(0.0 - d) ^ 0.5) / (Float64(0.0 - l) ^ 0.5)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(h * Float64(Float64(-0.5 / t_0) / l))) / Float64(d * 4.0)))));
	elseif (d <= 1.05e-216)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 6.8e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * t_0)) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 7.5e+169)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)))) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d / (M_m * D_m);
	tmp = 0.0;
	if (d <= -7.8e-230)
		tmp = (((0.0 - d) ^ 0.5) / ((0.0 - l) ^ 0.5)) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0))));
	elseif (d <= 1.05e-216)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 6.8e-79)
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 7.5e+169)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (sqrt(d) * (sqrt((d / l)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.8e-230], N[(N[(N[Power[N[(0.0 - d), $MachinePrecision], 0.5], $MachinePrecision] / N[Power[N[(0.0 - l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h * N[(N[(-0.5 / t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-216], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+169], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -7.8000000000000004e-230

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 67.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 67.0%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.8%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 73.9%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{neg}\left(d\right)}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(\mathsf{neg}\left(d\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(d\right)\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(0 - d\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{d}, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{d}, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       8. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \left({\left(\mathsf{neg}\left(\ell\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       10. neg-sub0 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(0 - \ell\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

       11. --lowering--.f64 85.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \frac{1}{2}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 85.2%

       \[\leadsto \color{blue}{\frac{{\left(0 - d\right)}^{0.5}}{{\left(0 - \ell\right)}^{0.5}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)\right) \]

   if -7.8000000000000004e-230 < d < 1.0500000000000001e-216

   1. Initial program 26.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 13.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 10.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 75.6%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 1.0500000000000001e-216 < d < 6.79999999999999951e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 6.79999999999999951e-79 < d < 7.49999999999999992e169

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 68.5%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 7.49999999999999992e169 < d

   1. Initial program 83.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{{\left(0 - \ell\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell}\right)}{d \cdot 4}\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-216}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{M\_m \cdot D\_m}{d}\\ \mathbf{if}\;d \leq -4.1 \cdot 10^{-69}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 + \left(\frac{M\_m \cdot D\_m}{d \cdot 4} \cdot t\_1\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{\sqrt{0 - \ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(t\_1 \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right)\\ \mathbf{elif}\;d \leq 7.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(t\_0 \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (/ (* M_m D_m) d)))
   (if (<= d -4.1e-69)
     (*
      t_0
      (*
       (/ (sqrt (- 0.0 d)) (sqrt (- 0.0 h)))
       (+ 1.0 (* (* (/ (* M_m D_m) (* d 4.0)) t_1) (* -0.5 (/ h l))))))
     (if (<= d -2.8e-232)
       (*
        (/ (pow (- 0.0 d) 0.5) (sqrt (- 0.0 l)))
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* (* M_m D_m) (* t_1 (/ -0.5 (/ l h)))) (* d 4.0)))))
       (if (<= d 7.4e-214)
         (/
          -1.0
          (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
         (if (<= d 5.8e-79)
           (*
            (+
             1.0
             (/
              (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
              (* d 4.0)))
            (* d (pow (* l h) -0.5)))
           (if (<= d 1.36e+169)
             (*
              (/ d (sqrt h))
              (/
               (+
                1.0
                (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
               (sqrt l)))
             (/
              (*
               (sqrt d)
               (*
                t_0
                (-
                 1.0
                 (/
                  (*
                   (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d)))
                   (* 0.5 h))
                  l))))
              (sqrt h)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (M_m * D_m) / d;
	double tmp;
	if (d <= -4.1e-69) {
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * t_1) * (-0.5 * (h / l)))));
	} else if (d <= -2.8e-232) {
		tmp = (pow((0.0 - d), 0.5) / sqrt((0.0 - l))) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (t_1 * (-0.5 / (l / h)))) / (d * 4.0))));
	} else if (d <= 7.4e-214) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 1.36e+169) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (m_m * d_m) / d
    if (d <= (-4.1d-69)) then
        tmp = t_0 * ((sqrt((0.0d0 - d)) / sqrt((0.0d0 - h))) * (1.0d0 + ((((m_m * d_m) / (d * 4.0d0)) * t_1) * ((-0.5d0) * (h / l)))))
    else if (d <= (-2.8d-232)) then
        tmp = (((0.0d0 - d) ** 0.5d0) / sqrt((0.0d0 - l))) * (sqrt((d / h)) * (1.0d0 + (((m_m * d_m) * (t_1 * ((-0.5d0) / (l / h)))) / (d * 4.0d0))))
    else if (d <= 7.4d-214) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.8d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 1.36d+169) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (sqrt(d) * (t_0 * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)))) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (M_m * D_m) / d;
	double tmp;
	if (d <= -4.1e-69) {
		tmp = t_0 * ((Math.sqrt((0.0 - d)) / Math.sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * t_1) * (-0.5 * (h / l)))));
	} else if (d <= -2.8e-232) {
		tmp = (Math.pow((0.0 - d), 0.5) / Math.sqrt((0.0 - l))) * (Math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (t_1 * (-0.5 / (l / h)))) / (d * 4.0))));
	} else if (d <= 7.4e-214) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 1.36e+169) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (M_m * D_m) / d
	tmp = 0
	if d <= -4.1e-69:
		tmp = t_0 * ((math.sqrt((0.0 - d)) / math.sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * t_1) * (-0.5 * (h / l)))))
	elif d <= -2.8e-232:
		tmp = (math.pow((0.0 - d), 0.5) / math.sqrt((0.0 - l))) * (math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (t_1 * (-0.5 / (l / h)))) / (d * 4.0))))
	elif d <= 7.4e-214:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.8e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 1.36e+169:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (math.sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M_m * D_m) / d)
	tmp = 0.0
	if (d <= -4.1e-69)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(0.0 - d)) / sqrt(Float64(0.0 - h))) * Float64(1.0 + Float64(Float64(Float64(Float64(M_m * D_m) / Float64(d * 4.0)) * t_1) * Float64(-0.5 * Float64(h / l))))));
	elseif (d <= -2.8e-232)
		tmp = Float64(Float64((Float64(0.0 - d) ^ 0.5) / sqrt(Float64(0.0 - l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(t_1 * Float64(-0.5 / Float64(l / h)))) / Float64(d * 4.0)))));
	elseif (d <= 7.4e-214)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.8e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 1.36e+169)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)))) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (M_m * D_m) / d;
	tmp = 0.0;
	if (d <= -4.1e-69)
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * t_1) * (-0.5 * (h / l)))));
	elseif (d <= -2.8e-232)
		tmp = (((0.0 - d) ^ 0.5) / sqrt((0.0 - l))) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (t_1 * (-0.5 / (l / h)))) / (d * 4.0))));
	elseif (d <= 7.4e-214)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.8e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 1.36e+169)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.1e-69], N[(t$95$0 * N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-232], N[(N[(N[Power[N[(0.0 - d), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(t$95$1 * N[(-0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.4e-214], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.36e+169], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -4.0999999999999999e-69

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 74.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{1}{h}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{-1}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot -1}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot -1}}{\sqrt{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{d \cdot -1}\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(d \cdot -1\right)\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\left(0 - h\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 84.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, h\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 84.9%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d \cdot -1}}{\sqrt{0 - h}}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   if -4.0999999999999999e-69 < d < -2.79999999999999993e-232

   1. Initial program 51.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 51.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 51.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 54.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 54.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left(\mathsf{neg}\left(d\right)\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(\mathsf{neg}\left(d\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(d\right)\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(d, h\right)}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(0 - d\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{d}, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{d}, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(0 - \ell\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 71.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{{\left(0 - d\right)}^{0.5}}{\sqrt{0 - \ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right) \]

   if -2.79999999999999993e-232 < d < 7.4000000000000004e-214

   1. Initial program 26.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 13.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 10.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 75.6%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 7.4000000000000004e-214 < d < 5.8000000000000001e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.8000000000000001e-79 < d < 1.36000000000000001e169

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 68.5%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 1.36000000000000001e169 < d

   1. Initial program 83.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{\sqrt{0 - \ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right)\\ \mathbf{elif}\;d \leq 7.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-215}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 + \left(\frac{M\_m \cdot D\_m}{d \cdot 4} \cdot \frac{M\_m \cdot D\_m}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(t\_0 \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -2.1e-215)
     (*
      t_0
      (*
       (/ (sqrt (- 0.0 d)) (sqrt (- 0.0 h)))
       (+
        1.0
        (* (* (/ (* M_m D_m) (* d 4.0)) (/ (* M_m D_m) d)) (* -0.5 (/ h l))))))
     (if (<= d 2.5e-213)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.2e-79)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 1.7e+169)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           (/
            (*
             (sqrt d)
             (*
              t_0
              (-
               1.0
               (/
                (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
                l))))
            (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -2.1e-215) {
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	} else if (d <= 2.5e-213) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.2e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 1.7e+169) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (d <= (-2.1d-215)) then
        tmp = t_0 * ((sqrt((0.0d0 - d)) / sqrt((0.0d0 - h))) * (1.0d0 + ((((m_m * d_m) / (d * 4.0d0)) * ((m_m * d_m) / d)) * ((-0.5d0) * (h / l)))))
    else if (d <= 2.5d-213) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.2d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 1.7d+169) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (sqrt(d) * (t_0 * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)))) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (d <= -2.1e-215) {
		tmp = t_0 * ((Math.sqrt((0.0 - d)) / Math.sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	} else if (d <= 2.5e-213) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.2e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 1.7e+169) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -2.1e-215:
		tmp = t_0 * ((math.sqrt((0.0 - d)) / math.sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))))
	elif d <= 2.5e-213:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.2e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 1.7e+169:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (math.sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -2.1e-215)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(0.0 - d)) / sqrt(Float64(0.0 - h))) * Float64(1.0 + Float64(Float64(Float64(Float64(M_m * D_m) / Float64(d * 4.0)) * Float64(Float64(M_m * D_m) / d)) * Float64(-0.5 * Float64(h / l))))));
	elseif (d <= 2.5e-213)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.2e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 1.7e+169)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)))) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -2.1e-215)
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	elseif (d <= 2.5e-213)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.2e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 1.7e+169)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (sqrt(d) * (t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.1e-215], N[(t$95$0 * N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e-213], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+169], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.1e-215

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{1}{h}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{-1}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot -1}{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot -1}}{\sqrt{\mathsf{neg}\left(h\right)}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{d \cdot -1}\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(d \cdot -1\right)\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \left(\sqrt{\mathsf{neg}\left(h\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(h\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\left(0 - h\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 77.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(d, -1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, h\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 77.8%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d \cdot -1}}{\sqrt{0 - h}}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   if -2.1e-215 < d < 2.49999999999999989e-213

   1. Initial program 26.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 12.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 11.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 71.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.49999999999999989e-213 < d < 5.19999999999999987e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.19999999999999987e-79 < d < 1.70000000000000014e169

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 68.5%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 1.70000000000000014e169 < d

   1. Initial program 83.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{M\_m \cdot D\_m}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(h \cdot \frac{\frac{-0.5}{t\_0}}{\ell}\right)}{d \cdot 4}\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot t\_0} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(t\_1 \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (* M_m D_m))) (t_1 (sqrt (/ d l))))
   (if (<= d -1.5e-214)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* M_m D_m) (* h (/ (/ -0.5 t_0) l))) (* d 4.0))))
      t_1)
     (if (<= d 3.6e-218)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 6.8e-79)
         (*
          (+ 1.0 (/ (* (/ -0.5 (* l t_0)) (* h (* M_m D_m))) (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 1.4e+169)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           (/
            (*
             (sqrt d)
             (*
              t_1
              (-
               1.0
               (/
                (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
                l))))
            (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.5e-214) {
		tmp = (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * t_1;
	} else if (d <= 3.6e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 6.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 1.4e+169) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (sqrt(d) * (t_1 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / (m_m * d_m)
    t_1 = sqrt((d / l))
    if (d <= (-1.5d-214)) then
        tmp = (sqrt((d / h)) * (1.0d0 + (((m_m * d_m) * (h * (((-0.5d0) / t_0) / l))) / (d * 4.0d0)))) * t_1
    else if (d <= 3.6d-218) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 6.8d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * t_0)) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 1.4d+169) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (sqrt(d) * (t_1 * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)))) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.5e-214) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * t_1;
	} else if (d <= 3.6e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 6.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 1.4e+169) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(d) * (t_1 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d / (M_m * D_m)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.5e-214:
		tmp = (math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * t_1
	elif d <= 3.6e-218:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 6.8e-79:
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 1.4e+169:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (math.sqrt(d) * (t_1 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / Float64(M_m * D_m))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.5e-214)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(h * Float64(Float64(-0.5 / t_0) / l))) / Float64(d * 4.0)))) * t_1);
	elseif (d <= 3.6e-218)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 6.8e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * t_0)) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 1.4e+169)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)))) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d / (M_m * D_m);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.5e-214)
		tmp = (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * t_1;
	elseif (d <= 3.6e-218)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 6.8e-79)
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 1.4e+169)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (sqrt(d) * (t_1 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)))) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.5e-214], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h * N[(N[(-0.5 / t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.6e-218], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+169], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.49999999999999997e-214

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 68.9%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 75.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 75.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]

   if -1.49999999999999997e-214 < d < 3.60000000000000011e-218

   1. Initial program 26.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 12.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 11.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 71.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 3.60000000000000011e-218 < d < 6.79999999999999951e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 6.79999999999999951e-79 < d < 1.4000000000000001e169

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 68.5%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 1.4000000000000001e169 < d

   1. Initial program 83.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell}\right)}{d \cdot 4}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{M\_m \cdot D\_m}\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(h \cdot \frac{\frac{-0.5}{t\_0}}{\ell}\right)}{d \cdot 4}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -4 \cdot 10^{-212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot t\_0} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (* M_m D_m)))
        (t_1
         (*
          (*
           (sqrt (/ d h))
           (+ 1.0 (/ (* (* M_m D_m) (* h (/ (/ -0.5 t_0) l))) (* d 4.0))))
          (sqrt (/ d l)))))
   (if (<= d -4e-212)
     t_1
     (if (<= d 2.5e-211)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.2e-79)
         (*
          (+ 1.0 (/ (* (/ -0.5 (* l t_0)) (* h (* M_m D_m))) (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 3.8e+157)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           t_1))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double t_1 = (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * sqrt((d / l));
	double tmp;
	if (d <= -4e-212) {
		tmp = t_1;
	} else if (d <= 2.5e-211) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.2e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 3.8e+157) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / (m_m * d_m)
    t_1 = (sqrt((d / h)) * (1.0d0 + (((m_m * d_m) * (h * (((-0.5d0) / t_0) / l))) / (d * 4.0d0)))) * sqrt((d / l))
    if (d <= (-4d-212)) then
        tmp = t_1
    else if (d <= 2.5d-211) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.2d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * t_0)) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 3.8d+157) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / (M_m * D_m);
	double t_1 = (Math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * Math.sqrt((d / l));
	double tmp;
	if (d <= -4e-212) {
		tmp = t_1;
	} else if (d <= 2.5e-211) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.2e-79) {
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 3.8e+157) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d / (M_m * D_m)
	t_1 = (math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * math.sqrt((d / l))
	tmp = 0
	if d <= -4e-212:
		tmp = t_1
	elif d <= 2.5e-211:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.2e-79:
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 3.8e+157:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = t_1
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / Float64(M_m * D_m))
	t_1 = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(h * Float64(Float64(-0.5 / t_0) / l))) / Float64(d * 4.0)))) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (d <= -4e-212)
		tmp = t_1;
	elseif (d <= 2.5e-211)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.2e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * t_0)) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 3.8e+157)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d / (M_m * D_m);
	t_1 = (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (h * ((-0.5 / t_0) / l))) / (d * 4.0)))) * sqrt((d / l));
	tmp = 0.0;
	if (d <= -4e-212)
		tmp = t_1;
	elseif (d <= 2.5e-211)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.2e-79)
		tmp = (1.0 + (((-0.5 / (l * t_0)) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 3.8e+157)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h * N[(N[(-0.5 / t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4e-212], t$95$1, If[LessEqual[d, 2.5e-211], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e+157], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.99999999999999982e-212 or 3.8000000000000001e157 < d

   1. Initial program 72.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 72.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 73.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 77.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]

   if -3.99999999999999982e-212 < d < 2.5000000000000001e-211

   1. Initial program 26.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 12.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 11.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 71.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.5000000000000001e-211 < d < 5.19999999999999987e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.19999999999999987e-79 < d < 3.8000000000000001e157

   1. Initial program 75.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 69.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-212}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell}\right)}{d \cdot 4}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell}\right)}{d \cdot 4}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right)\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (*
           (sqrt (/ d h))
           (+
            1.0
            (/
             (* (* M_m D_m) (* (/ (* M_m D_m) d) (/ -0.5 (/ l h))))
             (* d 4.0)))))))
   (if (<= d -1.85e-230)
     t_0
     (if (<= d 4e-217)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.5e-79)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 3.3e+158)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (((M_m * D_m) / d) * (-0.5 / (l / h)))) / (d * 4.0))));
	double tmp;
	if (d <= -1.85e-230) {
		tmp = t_0;
	} else if (d <= 4e-217) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 3.3e+158) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((m_m * d_m) * (((m_m * d_m) / d) * ((-0.5d0) / (l / h)))) / (d * 4.0d0))))
    if (d <= (-1.85d-230)) then
        tmp = t_0
    else if (d <= 4d-217) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.5d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 3.3d+158) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (((M_m * D_m) / d) * (-0.5 / (l / h)))) / (d * 4.0))));
	double tmp;
	if (d <= -1.85e-230) {
		tmp = t_0;
	} else if (d <= 4e-217) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 3.3e+158) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (((M_m * D_m) * (((M_m * D_m) / d) * (-0.5 / (l / h)))) / (d * 4.0))))
	tmp = 0
	if d <= -1.85e-230:
		tmp = t_0
	elif d <= 4e-217:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.5e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 3.3e+158:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) / d) * Float64(-0.5 / Float64(l / h)))) / Float64(d * 4.0)))))
	tmp = 0.0
	if (d <= -1.85e-230)
		tmp = t_0;
	elseif (d <= 4e-217)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.5e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 3.3e+158)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((M_m * D_m) * (((M_m * D_m) / d) * (-0.5 / (l / h)))) / (d * 4.0))));
	tmp = 0.0;
	if (d <= -1.85e-230)
		tmp = t_0;
	elseif (d <= 4e-217)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.5e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 3.3e+158)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e-230], t$95$0, If[LessEqual[d, 4e-217], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.3e+158], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.84999999999999991e-230 or 3.30000000000000017e158 < d

   1. Initial program 71.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 71.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 71.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 72.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]

   if -1.84999999999999991e-230 < d < 4.00000000000000033e-217

   1. Initial program 26.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 13.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 10.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 75.6%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 4.00000000000000033e-217 < d < 5.4999999999999997e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.4999999999999997e-79 < d < 3.30000000000000017e158

   1. Initial program 75.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 69.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(\frac{M\_m \cdot D\_m}{d \cdot 4} \cdot \frac{M\_m \cdot D\_m}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{-213}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+158}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (*
           (sqrt (/ d h))
           (+
            1.0
            (*
             (* (/ (* M_m D_m) (* d 4.0)) (/ (* M_m D_m) d))
             (* -0.5 (/ h l))))))))
   (if (<= d -1.2e-213)
     t_0
     (if (<= d 3.1e-218)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.8e-79)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 1.9e+158)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	double tmp;
	if (d <= -1.2e-213) {
		tmp = t_0;
	} else if (d <= 3.1e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 1.9e+158) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((((m_m * d_m) / (d * 4.0d0)) * ((m_m * d_m) / d)) * ((-0.5d0) * (h / l)))))
    if (d <= (-1.2d-213)) then
        tmp = t_0
    else if (d <= 3.1d-218) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.8d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 1.9d+158) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	double tmp;
	if (d <= -1.2e-213) {
		tmp = t_0;
	} else if (d <= 3.1e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 1.9e+158) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))))
	tmp = 0
	if d <= -1.2e-213:
		tmp = t_0
	elif d <= 3.1e-218:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.8e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 1.9e+158:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(Float64(M_m * D_m) / Float64(d * 4.0)) * Float64(Float64(M_m * D_m) / d)) * Float64(-0.5 * Float64(h / l))))))
	tmp = 0.0
	if (d <= -1.2e-213)
		tmp = t_0;
	elseif (d <= 3.1e-218)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.8e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 1.9e+158)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((((M_m * D_m) / (d * 4.0)) * ((M_m * D_m) / d)) * (-0.5 * (h / l)))));
	tmp = 0.0;
	if (d <= -1.2e-213)
		tmp = t_0;
	elseif (d <= 3.1e-218)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.8e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 1.9e+158)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.2e-213], t$95$0, If[LessEqual[d, 3.1e-218], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e+158], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.19999999999999998e-213 or 1.8999999999999999e158 < d

   1. Initial program 72.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 72.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   if -1.19999999999999998e-213 < d < 3.09999999999999997e-218

   1. Initial program 26.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 12.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 16.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 16.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 11.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 71.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 3.09999999999999997e-218 < d < 5.8000000000000001e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.8000000000000001e-79 < d < 1.8999999999999999e158

   1. Initial program 75.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 69.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+158}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(M\_m \cdot D\_m\right)\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D\_m}}{t\_0}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot t\_0}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* M_m D_m))))
   (if (<= d -2.5e-126)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* h (/ -0.125 (/ (/ (* d d) D_m) t_0))) l))))
     (if (<= d 3.5e-218)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.5e-79)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 7.5e+109)
           (*
            (/ d (sqrt h))
            (/
             (+ 1.0 (/ (/ (* h -0.125) (/ (* d d) (* D_m t_0))) l))
             (sqrt l)))
           (*
            (-
             1.0
             (/
              (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
              l))
            (/ d (pow (* l h) 0.5)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (M_m * D_m);
	double tmp;
	if (d <= -2.5e-126) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.125 / (((d * d) / D_m) / t_0))) / l)));
	} else if (d <= 3.5e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 7.5e+109) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * t_0))) / l)) / sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / pow((l * h), 0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (m_m * d_m)
    if (d <= (-2.5d-126)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.125d0) / (((d * d) / d_m) / t_0))) / l)))
    else if (d <= 3.5d-218) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.5d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 7.5d+109) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * t_0))) / l)) / sqrt(l))
    else
        tmp = (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)) * (d / ((l * h) ** 0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (M_m * D_m);
	double tmp;
	if (d <= -2.5e-126) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.125 / (((d * d) / D_m) / t_0))) / l)));
	} else if (d <= 3.5e-218) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 7.5e+109) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * t_0))) / l)) / Math.sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / Math.pow((l * h), 0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * (M_m * D_m)
	tmp = 0
	if d <= -2.5e-126:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * (-0.125 / (((d * d) / D_m) / t_0))) / l)))
	elif d <= 3.5e-218:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.5e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 7.5e+109:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * t_0))) / l)) / math.sqrt(l))
	else:
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / math.pow((l * h), 0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(M_m * D_m))
	tmp = 0.0
	if (d <= -2.5e-126)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.125 / Float64(Float64(Float64(d * d) / D_m) / t_0))) / l))));
	elseif (d <= 3.5e-218)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.5e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 7.5e+109)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * t_0))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)) * Float64(d / (Float64(l * h) ^ 0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * (M_m * D_m);
	tmp = 0.0;
	if (d <= -2.5e-126)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.125 / (((d * d) / D_m) / t_0))) / l)));
	elseif (d <= 3.5e-218)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.5e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 7.5e+109)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * t_0))) / l)) / sqrt(l));
	else
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / ((l * h) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e-126], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.125 / N[(N[(N[(d * d), $MachinePrecision] / D$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-218], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+109], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.50000000000000003e-126

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 71.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 71.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   8. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -2.50000000000000003e-126 < d < 3.5e-218

   1. Initial program 34.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 18.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 9.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 21.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 21.3%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 8.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 63.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 63.1%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 3.5e-218 < d < 5.4999999999999997e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.4999999999999997e-79 < d < 7.50000000000000018e109

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 78.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 7.50000000000000018e109 < d

   1. Initial program 82.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(-0.125 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{h \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.6e-58)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (* -0.125 (* M_m M_m)) (/ (* h (/ (/ (* D_m D_m) d) d)) l)))))
   (if (<= d 6.6e-215)
     (/
      -1.0
      (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
     (if (<= d 5.8e-79)
       (*
        (+
         1.0
         (/ (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m))) (* d 4.0)))
        (* d (pow (* l h) -0.5)))
       (if (<= d 5e+111)
         (*
          (/ d (sqrt h))
          (/
           (+
            1.0
            (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
           (sqrt l)))
         (*
          (-
           1.0
           (/
            (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
            l))
          (/ d (pow (* l h) 0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.6e-58) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((-0.125 * (M_m * M_m)) * ((h * (((D_m * D_m) / d) / d)) / l))));
	} else if (d <= 6.6e-215) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 5e+111) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / pow((l * h), 0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.6d-58)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((-0.125d0) * (m_m * m_m)) * ((h * (((d_m * d_m) / d) / d)) / l))))
    else if (d <= 6.6d-215) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.8d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 5d+111) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)) * (d / ((l * h) ** 0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.6e-58) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((-0.125 * (M_m * M_m)) * ((h * (((D_m * D_m) / d) / d)) / l))));
	} else if (d <= 6.6e-215) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.8e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 5e+111) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / Math.pow((l * h), 0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2.6e-58:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((-0.125 * (M_m * M_m)) * ((h * (((D_m * D_m) / d) / d)) / l))))
	elif d <= 6.6e-215:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.8e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 5e+111:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / math.pow((l * h), 0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.6e-58)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(-0.125 * Float64(M_m * M_m)) * Float64(Float64(h * Float64(Float64(Float64(D_m * D_m) / d) / d)) / l)))));
	elseif (d <= 6.6e-215)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.8e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 5e+111)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)) * Float64(d / (Float64(l * h) ^ 0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2.6e-58)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((-0.125 * (M_m * M_m)) * ((h * (((D_m * D_m) / d) / d)) / l))));
	elseif (d <= 6.6e-215)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.8e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 5e+111)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / ((l * h) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.6e-58], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e-215], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+111], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.60000000000000007e-58

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 64.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 74.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 74.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 74.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 80.9%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]

   11. Taylor expanded in M around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} + \frac{1}{{M}^{2}}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left({M}^{2} \cdot \left(\frac{1}{{M}^{2}} + \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left({M}^{2} \cdot \frac{1}{{M}^{2}} + \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

       3. rgt-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{{M}^{2}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)}\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2} \cdot \frac{-1}{8}\right), \color{blue}{\left(\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{8}\right), \left(\frac{\color{blue}{{D}^{2} \cdot h}}{{d}^{2} \cdot \ell}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{8}\right), \left(\frac{\color{blue}{{D}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{8}\right), \left(\frac{\color{blue}{{D}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right)\right) \]

       10. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{8}\right), \left(\frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\color{blue}{\ell}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{8}\right), \mathsf{/.f64}\left(\left(\frac{{D}^{2} \cdot h}{{d}^{2}}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   13. Simplified 64.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(1 + \left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{h \cdot \frac{\frac{D \cdot D}{d}}{d}}{\ell}\right)}\right) \]

   if -2.60000000000000007e-58 < d < 6.5999999999999996e-215

   1. Initial program 41.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 28.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 28.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 6.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 61.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 61.3%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 6.5999999999999996e-215 < d < 5.8000000000000001e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.8000000000000001e-79 < d < 4.9999999999999997e111

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 78.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 4.9999999999999997e111 < d

   1. Initial program 82.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{h \cdot \frac{\frac{D \cdot D}{d}}{d}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)\\ \mathbf{if}\;d \leq -8 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + t\_0 \cdot \left(-0.125 \cdot \frac{\frac{h}{\ell}}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{t\_0} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (* M_m D_m)))))
   (if (<= d -8e-61)
     (*
      (sqrt (/ d l))
      (* (sqrt (/ d h)) (+ 1.0 (* t_0 (* -0.125 (/ (/ h l) (* d d)))))))
     (if (<= d 4.4e-218)
       (/ -1.0 (* (/ d t_0) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= d 5.5e-79)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (if (<= d 5e+111)
           (*
            (/ d (sqrt h))
            (/
             (+
              1.0
              (/ (/ (* h -0.125) (/ (* d d) (* D_m (* M_m (* M_m D_m))))) l))
             (sqrt l)))
           (*
            (-
             1.0
             (/
              (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h))
              l))
            (/ d (pow (* l h) 0.5)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double tmp;
	if (d <= -8e-61) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (t_0 * (-0.125 * ((h / l) / (d * d))))));
	} else if (d <= 4.4e-218) {
		tmp = -1.0 / ((d / t_0) * (8.0 * (l * sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else if (d <= 5e+111) {
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / pow((l * h), 0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_m * (m_m * d_m))
    if (d <= (-8d-61)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (t_0 * ((-0.125d0) * ((h / l) / (d * d))))))
    else if (d <= 4.4d-218) then
        tmp = (-1.0d0) / ((d / t_0) * (8.0d0 * (l * sqrt((l / h)))))
    else if (d <= 5.5d-79) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else if (d <= 5d+111) then
        tmp = (d / sqrt(h)) * ((1.0d0 + (((h * (-0.125d0)) / ((d * d) / (d_m * (m_m * (m_m * d_m))))) / l)) / sqrt(l))
    else
        tmp = (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l)) * (d / ((l * h) ** 0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m * (D_m * (M_m * D_m));
	double tmp;
	if (d <= -8e-61) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (t_0 * (-0.125 * ((h / l) / (d * d))))));
	} else if (d <= 4.4e-218) {
		tmp = -1.0 / ((d / t_0) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (d <= 5.5e-79) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else if (d <= 5e+111) {
		tmp = (d / Math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / Math.sqrt(l));
	} else {
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / Math.pow((l * h), 0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m * (D_m * (M_m * D_m))
	tmp = 0
	if d <= -8e-61:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (t_0 * (-0.125 * ((h / l) / (d * d))))))
	elif d <= 4.4e-218:
		tmp = -1.0 / ((d / t_0) * (8.0 * (l * math.sqrt((l / h)))))
	elif d <= 5.5e-79:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	elif d <= 5e+111:
		tmp = (d / math.sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / math.sqrt(l))
	else:
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / math.pow((l * h), 0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(M_m * D_m)))
	tmp = 0.0
	if (d <= -8e-61)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(t_0 * Float64(-0.125 * Float64(Float64(h / l) / Float64(d * d)))))));
	elseif (d <= 4.4e-218)
		tmp = Float64(-1.0 / Float64(Float64(d / t_0) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (d <= 5.5e-79)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (d <= 5e+111)
		tmp = Float64(Float64(d / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.125) / Float64(Float64(d * d) / Float64(D_m * Float64(M_m * Float64(M_m * D_m))))) / l)) / sqrt(l)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(Float64(D_m / d) * Float64(D_m / d))) * Float64(0.5 * h)) / l)) * Float64(d / (Float64(l * h) ^ 0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m * (D_m * (M_m * D_m));
	tmp = 0.0;
	if (d <= -8e-61)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (t_0 * (-0.125 * ((h / l) / (d * d))))));
	elseif (d <= 4.4e-218)
		tmp = -1.0 / ((d / t_0) * (8.0 * (l * sqrt((l / h)))));
	elseif (d <= 5.5e-79)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	elseif (d <= 5e+111)
		tmp = (d / sqrt(h)) * ((1.0 + (((h * -0.125) / ((d * d) / (D_m * (M_m * (M_m * D_m))))) / l)) / sqrt(l));
	else
		tmp = (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l)) * (d / ((l * h) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8e-61], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-218], N[(-1.0 / N[(N[(d / t$95$0), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e-79], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+111], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] / N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -8.0000000000000003e-61

   1. Initial program 74.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 74.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 74.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)}\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot 4\right) \cdot d}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{\left(D \cdot M\right) \cdot \left(M \cdot D\right)}{\left(\color{blue}{d} \cdot 4\right) \cdot d}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(d \cdot 4\right)} \cdot d}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\left(4 \cdot d\right) \cdot d}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{4 \cdot \color{blue}{\left(d \cdot d\right)}}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{h}{\ell} \cdot \frac{-1}{2}\right) \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\color{blue}{4 \cdot \left(d \cdot d\right)}}\right)\right)\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{h}{\ell} \cdot \frac{-1}{2}}{4 \cdot \left(d \cdot d\right)} \cdot \color{blue}{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \color{blue}{\frac{\frac{h}{\ell} \cdot \frac{-1}{2}}{4 \cdot \left(d \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right), \color{blue}{\left(\frac{\frac{h}{\ell} \cdot \frac{-1}{2}}{4 \cdot \left(d \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(D \cdot M\right) \cdot \left(M \cdot D\right)\right), \left(\frac{\color{blue}{\frac{h}{\ell} \cdot \frac{-1}{2}}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), \left(\frac{\color{blue}{\frac{h}{\ell}} \cdot \frac{-1}{2}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right), \left(\frac{\color{blue}{\frac{h}{\ell} \cdot \frac{-1}{2}}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right), \left(\frac{\color{blue}{\frac{h}{\ell} \cdot \frac{-1}{2}}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{h}{\ell} \cdot \color{blue}{\frac{-1}{2}}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{h}{\ell} \cdot \frac{-1}{2}}{4 \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{-1}{2} \cdot \frac{h}{\ell}}{\color{blue}{4} \cdot \left(d \cdot d\right)}\right)\right)\right)\right)\right) \]

      18. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{-1}{2}}{4} \cdot \color{blue}{\frac{\frac{h}{\ell}}{d \cdot d}}\right)\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{\color{blue}{\frac{h}{\ell}}}{d \cdot d}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 62.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(-0.125 \cdot \frac{\frac{h}{\ell}}{d \cdot d}\right)}\right)\right) \]

   if -8.0000000000000003e-61 < d < 4.40000000000000014e-218

   1. Initial program 40.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 29.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 28.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 6.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 62.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 62.1%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 4.40000000000000014e-218 < d < 5.4999999999999997e-79

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 5.4999999999999997e-79 < d < 4.9999999999999997e111

   1. Initial program 75.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   8. Applied egg-rr 78.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{-0.125}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}} \cdot \left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right)}{\color{blue}{\sqrt{\ell}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{d \cdot \frac{d}{h}} \cdot \color{blue}{\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\frac{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\frac{\color{blue}{1} + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}{\sqrt{\ell}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\frac{\color{blue}{1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\frac{1 + \color{blue}{\frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}}}{\sqrt{\ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{h \cdot \frac{\frac{-1}{8}}{\frac{\frac{d \cdot d}{D}}{M \cdot \left(M \cdot D\right)}}}{\ell}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right)\right) \]

   10. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}} \]

   if 4.9999999999999997e111 < d

   1. Initial program 82.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \left(-0.125 \cdot \frac{\frac{h}{\ell}}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1 + \frac{\frac{h \cdot -0.125}{\frac{d \cdot d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}}{\ell}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right) \cdot \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{{\left(\frac{\ell}{d}\right)}^{0.5}}\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -9.4e+222)
   (/ (sqrt (/ d h)) (pow (/ l d) 0.5))
   (if (<= l -1.7e+75)
     (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
     (if (<= l 2.5e-285)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= l 2.25e+169)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (* d (/ (pow l -0.5) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.4e+222) {
		tmp = sqrt((d / h)) / pow((l / d), 0.5);
	} else if (l <= -1.7e+75) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (l <= 2.25e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-9.4d+222)) then
        tmp = sqrt((d / h)) / ((l / d) ** 0.5d0)
    else if (l <= (-1.7d+75)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (l <= 2.5d-285) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (l <= 2.25d+169) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.4e+222) {
		tmp = Math.sqrt((d / h)) / Math.pow((l / d), 0.5);
	} else if (l <= -1.7e+75) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (l <= 2.25e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -9.4e+222:
		tmp = math.sqrt((d / h)) / math.pow((l / d), 0.5)
	elif l <= -1.7e+75:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif l <= 2.5e-285:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif l <= 2.25e+169:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -9.4e+222)
		tmp = Float64(sqrt(Float64(d / h)) / (Float64(l / d) ^ 0.5));
	elseif (l <= -1.7e+75)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (l <= 2.5e-285)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (l <= 2.25e+169)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -9.4e+222)
		tmp = sqrt((d / h)) / ((l / d) ^ 0.5);
	elseif (l <= -1.7e+75)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (l <= 2.5e-285)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (l <= 2.25e+169)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -9.4e+222], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] / N[Power[N[(l / d), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.7e+75], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-285], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+169], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -9.3999999999999998e222

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 10.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 10.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 10.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 10.1%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{1 \cdot \frac{d}{h}}{\frac{\ell}{d}}} \]

       10. clear-num N/A

           \[\leadsto \sqrt{\frac{1 \cdot \frac{1}{\frac{h}{d}}}{\frac{\ell}{d}}} \]

       11. div-inv N/A

           \[\leadsto \sqrt{\frac{\frac{1}{\frac{h}{d}}}{\frac{\ell}{d}}} \]

       12. clear-num N/A

           \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \]

       13. sqrt-div N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \]

       14. sqrt-div N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{d}}}} \]

       15. pow1/2 N/A

           \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\frac{\sqrt{\ell}}{{d}^{\color{blue}{\frac{1}{2}}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \color{blue}{\left(\frac{\sqrt{\ell}}{{d}^{\frac{1}{2}}}\right)}\right) \]

       17. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\frac{\color{blue}{\sqrt{\ell}}}{{d}^{\frac{1}{2}}}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(\frac{\sqrt{\color{blue}{\ell}}}{{d}^{\frac{1}{2}}}\right)\right) \]

       19. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(\frac{\sqrt{\ell}}{\sqrt{d}}\right)\right) \]

       20. sqrt-div N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(\sqrt{\frac{\ell}{d}}\right)\right) \]

       21. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left({\left(\frac{\ell}{d}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

       22. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{pow.f64}\left(\left(\frac{\ell}{d}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

       23. /-lowering-/.f64 58.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \frac{1}{2}\right)\right) \]

   11. Applied egg-rr 58.7%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{{\left(\frac{\ell}{d}\right)}^{0.5}}} \]

   if -9.3999999999999998e222 < l < -1.70000000000000006e75

   1. Initial program 45.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 45.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 45.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 55.4%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.70000000000000006e75 < l < 2.50000000000000009e-285

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 50.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 3.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 57.5%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.50000000000000009e-285 < l < 2.25e169

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 73.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 77.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 2.25e169 < l

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\sqrt{\frac{1}{\ell}}}{\color{blue}{\sqrt{h}}}\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{\sqrt{1}}{\sqrt{\ell}}}{\sqrt{\color{blue}{h}}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\ell}}\right), \color{blue}{\left(\sqrt{h}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{{\ell}^{\frac{1}{2}}}\right), \left(\sqrt{h}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\frac{-1}{2}}\right), \left(\sqrt{h}\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(h\right)\right)\right) \]

   9. Applied egg-rr 81.0%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.4 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{{\left(\frac{\ell}{d}\right)}^{0.5}}\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -9.5e+222)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l -2.05e+75)
     (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
     (if (<= l 2.5e-285)
       (/
        -1.0
        (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
       (if (<= l 2e+169)
         (*
          (+
           1.0
           (/
            (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m)))
            (* d 4.0)))
          (* d (pow (* l h) -0.5)))
         (* d (/ (pow l -0.5) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.5e+222) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -2.05e+75) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (l <= 2e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-9.5d+222)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= (-2.05d+75)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (l <= 2.5d-285) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (l <= 2d+169) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.5e+222) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -2.05e+75) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (l <= 2e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -9.5e+222:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= -2.05e+75:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif l <= 2.5e-285:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif l <= 2e+169:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -9.5e+222)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -2.05e+75)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (l <= 2.5e-285)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (l <= 2e+169)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -9.5e+222)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= -2.05e+75)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (l <= 2.5e-285)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (l <= 2e+169)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -9.5e+222], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.05e+75], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-285], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e+169], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -9.5000000000000001e222

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right) \]

      2. /-lowering-/.f64 58.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   if -9.5000000000000001e222 < l < -2.0499999999999999e75

   1. Initial program 45.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 45.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 45.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 55.4%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -2.0499999999999999e75 < l < 2.50000000000000009e-285

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 50.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 3.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 57.5%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.50000000000000009e-285 < l < 1.99999999999999987e169

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 73.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 77.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 1.99999999999999987e169 < l

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\sqrt{\frac{1}{\ell}}}{\color{blue}{\sqrt{h}}}\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{\sqrt{1}}{\sqrt{\ell}}}{\sqrt{\color{blue}{h}}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\ell}}\right), \color{blue}{\left(\sqrt{h}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{{\ell}^{\frac{1}{2}}}\right), \left(\sqrt{h}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\frac{-1}{2}}\right), \left(\sqrt{h}\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(h\right)\right)\right) \]

   9. Applied egg-rr 81.0%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -3.3e+75)
   (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
   (if (<= l 2.5e-285)
     (/
      -1.0
      (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
     (if (<= l 2.6e+169)
       (*
        (+
         1.0
         (/ (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m))) (* d 4.0)))
        (* d (pow (* l h) -0.5)))
       (* d (/ (pow l -0.5) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3.3e+75) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (l <= 2.6e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-3.3d+75)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (l <= 2.5d-285) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (l <= 2.6d+169) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -3.3e+75) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (l <= 2.6e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -3.3e+75:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif l <= 2.5e-285:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif l <= 2.6e+169:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -3.3e+75)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (l <= 2.5e-285)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (l <= 2.6e+169)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -3.3e+75)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (l <= 2.5e-285)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (l <= 2.6e+169)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -3.3e+75], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-285], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+169], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.29999999999999998e75

   1. Initial program 50.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 50.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 48.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -3.29999999999999998e75 < l < 2.50000000000000009e-285

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 50.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 3.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 57.5%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.50000000000000009e-285 < l < 2.6e169

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 73.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 77.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 2.6e169 < l

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\sqrt{\frac{1}{\ell}}}{\color{blue}{\sqrt{h}}}\right)\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{\sqrt{1}}{\sqrt{\ell}}}{\sqrt{\color{blue}{h}}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left(\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\ell}}\right), \color{blue}{\left(\sqrt{h}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left(\frac{1}{{\ell}^{\frac{1}{2}}}\right), \left(\sqrt{h}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\left({\ell}^{\frac{-1}{2}}\right), \left(\sqrt{h}\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \left(\sqrt{\color{blue}{h}}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\ell, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(h\right)\right)\right) \]

   9. Applied egg-rr 81.0%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.02e+76)
   (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
   (if (<= l 2.5e-285)
     (/
      -1.0
      (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
     (if (<= l 1.36e+169)
       (*
        (+
         1.0
         (/ (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m))) (* d 4.0)))
        (* d (pow (* l h) -0.5)))
       (/ (/ d (sqrt h)) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.02e+76) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else if (l <= 1.36e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.02d+76)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (l <= 2.5d-285) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else if (l <= 1.36d+169) then
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.02e+76) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (l <= 2.5e-285) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else if (l <= 1.36e+169) {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.02e+76:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif l <= 2.5e-285:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	elif l <= 1.36e+169:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.02e+76)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (l <= 2.5e-285)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	elseif (l <= 1.36e+169)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.02e+76)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (l <= 2.5e-285)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	elseif (l <= 1.36e+169)
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.02e+76], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-285], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.36e+169], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.02000000000000007e76

   1. Initial program 50.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 50.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 48.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.02000000000000007e76 < l < 2.50000000000000009e-285

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 50.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 3.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 57.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 57.5%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 2.50000000000000009e-285 < l < 1.36000000000000001e169

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 73.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 77.2%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if 1.36000000000000001e169 < l

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 42.3%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \]

       9. sqrt-div N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{h}}}{\color{blue}{\sqrt{\ell}}} \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{d \cdot \frac{d}{h}}\right), \color{blue}{\left(\sqrt{\ell}\right)}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d \cdot d}{h}}\right), \left(\sqrt{\ell}\right)\right) \]

       12. sqrt-div N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sqrt{d \cdot d}}{\sqrt{h}}\right), \left(\sqrt{\color{blue}{\ell}}\right)\right) \]

       13. sqrt-prod N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h}}\right), \left(\sqrt{\ell}\right)\right) \]

       14. rem-square-sqrt N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{d}{\sqrt{h}}\right), \left(\sqrt{\ell}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{h}\right)\right), \left(\sqrt{\color{blue}{\ell}}\right)\right) \]

       16. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \left(\sqrt{\ell}\right)\right) \]

       17. sqrt-lowering-sqrt.f64 78.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(h\right)\right), \mathsf{sqrt.f64}\left(\ell\right)\right) \]

   11. Applied egg-rr 78.4%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-285}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+169}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+74}:\\ \;\;\;\;t\_0 \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot M\_m\right) \cdot \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}}\right)\right) \cdot \frac{0.125}{d}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+134}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -7.4e+74)
     (* t_0 (- 0.0 d))
     (if (<= l -5e-310)
       (*
        (* (* D_m D_m) (* (* M_m M_m) (sqrt (/ (/ h (* l l)) l))))
        (/ 0.125 d))
       (if (<= l 3.8e-233)
         (* d (sqrt (/ 1.0 (* l h))))
         (if (<= l 2.5e-49)
           (*
            (sqrt (/ h (* l (* l l))))
            (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 d)))
           (if (<= l 4.2e+134)
             (* d t_0)
             (/ 1.0 (sqrt (* (/ l d) (/ h d)))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -7.4e+74) {
		tmp = t_0 * (0.0 - d);
	} else if (l <= -5e-310) {
		tmp = ((D_m * D_m) * ((M_m * M_m) * sqrt(((h / (l * l)) / l)))) * (0.125 / d);
	} else if (l <= 3.8e-233) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else if (l <= 2.5e-49) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	} else if (l <= 4.2e+134) {
		tmp = d * t_0;
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (l <= (-7.4d+74)) then
        tmp = t_0 * (0.0d0 - d)
    else if (l <= (-5d-310)) then
        tmp = ((d_m * d_m) * ((m_m * m_m) * sqrt(((h / (l * l)) / l)))) * (0.125d0 / d)
    else if (l <= 3.8d-233) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else if (l <= 2.5d-49) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / d))
    else if (l <= 4.2d+134) then
        tmp = d * t_0
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -7.4e+74) {
		tmp = t_0 * (0.0 - d);
	} else if (l <= -5e-310) {
		tmp = ((D_m * D_m) * ((M_m * M_m) * Math.sqrt(((h / (l * l)) / l)))) * (0.125 / d);
	} else if (l <= 3.8e-233) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= 2.5e-49) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	} else if (l <= 4.2e+134) {
		tmp = d * t_0;
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -7.4e+74:
		tmp = t_0 * (0.0 - d)
	elif l <= -5e-310:
		tmp = ((D_m * D_m) * ((M_m * M_m) * math.sqrt(((h / (l * l)) / l)))) * (0.125 / d)
	elif l <= 3.8e-233:
		tmp = d * math.sqrt((1.0 / (l * h)))
	elif l <= 2.5e-49:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d))
	elif l <= 4.2e+134:
		tmp = d * t_0
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -7.4e+74)
		tmp = Float64(t_0 * Float64(0.0 - d));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) * sqrt(Float64(Float64(h / Float64(l * l)) / l)))) * Float64(0.125 / d));
	elseif (l <= 3.8e-233)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= 2.5e-49)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / d)));
	elseif (l <= 4.2e+134)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -7.4e+74)
		tmp = t_0 * (0.0 - d);
	elseif (l <= -5e-310)
		tmp = ((D_m * D_m) * ((M_m * M_m) * sqrt(((h / (l * l)) / l)))) * (0.125 / d);
	elseif (l <= 3.8e-233)
		tmp = d * sqrt((1.0 / (l * h)));
	elseif (l <= 2.5e-49)
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	elseif (l <= 4.2e+134)
		tmp = d * t_0;
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -7.4e+74], N[(t$95$0 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e-233], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e-49], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+134], N[(d * t$95$0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -7.4000000000000002e74

   1. Initial program 50.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 50.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 48.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -7.4000000000000002e74 < l < -4.999999999999985e-310

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 51.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 51.2%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]

   10. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\frac{1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{1}{8}} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \cdot \frac{1}{8} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{1}{8}}{\color{blue}{d}} \]

       4. associate-/l* N/A

          \[\leadsto \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{\frac{1}{8}}{d}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{\frac{1}{8}}{d}\right)}\right) \]

   12. Simplified 52.6%

       \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}}\right)\right) \cdot \frac{0.125}{d}} \]

   if -4.999999999999985e-310 < l < 3.8e-233

   1. Initial program 46.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 37.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 61.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 61.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 61.2%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 61.2%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

   if 3.8e-233 < l < 2.4999999999999999e-49

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{d}}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{-1}{8}}{d}\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{\frac{\frac{-1}{8}}{d}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \color{blue}{\left(\frac{\frac{-1}{8}}{d}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right), \left(\frac{\color{blue}{\frac{-1}{8}}}{d}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot {M}^{2}\right)\right), \left(\frac{\color{blue}{\frac{-1}{8}}}{d}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2}\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      21. /-lowering-/.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \color{blue}{d}\right)\right)\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)} \]

   if 2.4999999999999999e-49 < l < 4.2000000000000002e134

   1. Initial program 81.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 63.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 4.2000000000000002e134 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 42.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 40.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 56.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}}\right)\right) \cdot \frac{0.125}{d}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M\_m \cdot D\_m}} \cdot \left(h \cdot \left(M\_m \cdot D\_m\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.58e-58)
   (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
   (if (<= d 3.6e-215)
     (/
      -1.0
      (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))
     (*
      (+
       1.0
       (/ (* (/ -0.5 (* l (/ d (* M_m D_m)))) (* h (* M_m D_m))) (* d 4.0)))
      (* d (pow (* l h) -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.58e-58) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (d <= 3.6e-215) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	} else {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * pow((l * h), -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.58d-58)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (d <= 3.6d-215) then
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    else
        tmp = (1.0d0 + ((((-0.5d0) / (l * (d / (m_m * d_m)))) * (h * (m_m * d_m))) / (d * 4.0d0))) * (d * ((l * h) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.58e-58) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (d <= 3.6e-215) {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	} else {
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * Math.pow((l * h), -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.58e-58:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif d <= 3.6e-215:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	else:
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * math.pow((l * h), -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.58e-58)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (d <= 3.6e-215)
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(l * Float64(d / Float64(M_m * D_m)))) * Float64(h * Float64(M_m * D_m))) / Float64(d * 4.0))) * Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.58e-58)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (d <= 3.6e-215)
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	else
		tmp = (1.0 + (((-0.5 / (l * (d / (M_m * D_m)))) * (h * (M_m * D_m))) / (d * 4.0))) * (d * ((l * h) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.58e-58], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e-215], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(-0.5 / N[(l * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.57999999999999997e-58

   1. Initial program 74.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 74.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 74.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 51.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.57999999999999997e-58 < d < 3.5999999999999999e-215

   1. Initial program 40.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 29.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 28.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 6.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 62.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 62.1%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]

   if 3.5999999999999999e-215 < d

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 76.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 77.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 77.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 81.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}{d \cdot 4}\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   12. Applied egg-rr 79.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5}{\ell \cdot \frac{d}{M \cdot D}} \cdot \left(h \cdot \left(M \cdot D\right)\right)}{d \cdot 4}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ t_1 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-162}:\\ \;\;\;\;t\_1 \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-233}:\\ \;\;\;\;d \cdot {\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+134}:\\ \;\;\;\;d \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))) (t_1 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5.6e-162)
     (* t_1 (- 0.0 d))
     (if (<= l 1.15e-233)
       (* d (pow (* t_0 t_0) 0.25))
       (if (<= l 2.2e-50)
         (*
          (sqrt (/ h (* l (* l l))))
          (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 d)))
         (if (<= l 7e+134) (* d t_1) (/ 1.0 (sqrt (* (/ l d) (/ h d))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.6e-162) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= 1.15e-233) {
		tmp = d * pow((t_0 * t_0), 0.25);
	} else if (l <= 2.2e-50) {
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	} else if (l <= 7e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (l * h)
    t_1 = sqrt(((1.0d0 / h) / l))
    if (l <= (-5.6d-162)) then
        tmp = t_1 * (0.0d0 - d)
    else if (l <= 1.15d-233) then
        tmp = d * ((t_0 * t_0) ** 0.25d0)
    else if (l <= 2.2d-50) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / d))
    else if (l <= 7d+134) then
        tmp = d * t_1
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.6e-162) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= 1.15e-233) {
		tmp = d * Math.pow((t_0 * t_0), 0.25);
	} else if (l <= 2.2e-50) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	} else if (l <= 7e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 / (l * h)
	t_1 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -5.6e-162:
		tmp = t_1 * (0.0 - d)
	elif l <= 1.15e-233:
		tmp = d * math.pow((t_0 * t_0), 0.25)
	elif l <= 2.2e-50:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d))
	elif l <= 7e+134:
		tmp = d * t_1
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 / Float64(l * h))
	t_1 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -5.6e-162)
		tmp = Float64(t_1 * Float64(0.0 - d));
	elseif (l <= 1.15e-233)
		tmp = Float64(d * (Float64(t_0 * t_0) ^ 0.25));
	elseif (l <= 2.2e-50)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / d)));
	elseif (l <= 7e+134)
		tmp = Float64(d * t_1);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 / (l * h);
	t_1 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -5.6e-162)
		tmp = t_1 * (0.0 - d);
	elseif (l <= 1.15e-233)
		tmp = d * ((t_0 * t_0) ^ 0.25);
	elseif (l <= 2.2e-50)
		tmp = sqrt((h / (l * (l * l)))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d));
	elseif (l <= 7e+134)
		tmp = d * t_1;
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.6e-162], N[(t$95$1 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e-233], N[(d * N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e-50], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e+134], N[(d * t$95$1), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -5.60000000000000043e-162

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -5.60000000000000043e-162 < l < 1.1500000000000001e-233

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 25.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 25.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{\color{blue}{4}}\right)}\right)\right) \]

      5. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)} \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\left(\frac{1}{4}\right)}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right), \color{blue}{\left(\frac{1}{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{\color{blue}{1}}{4}\right)\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell \cdot h}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{1}{\ell \cdot h}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      15. metadata-eval 48.6%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \frac{1}{4}\right)\right) \]

   9. Applied egg-rr 48.6%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}} \]

   if 1.1500000000000001e-233 < l < 2.1999999999999999e-50

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 56.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{d}}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{-1}{8}}{d}\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{\frac{\frac{-1}{8}}{d}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \color{blue}{\left(\frac{\frac{-1}{8}}{d}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right), \left(\frac{\color{blue}{\frac{-1}{8}}}{d}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot {M}^{2}\right)\right), \left(\frac{\color{blue}{\frac{-1}{8}}}{d}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2}\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right)\right) \]

      21. /-lowering-/.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \color{blue}{d}\right)\right)\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)} \]

   if 2.1999999999999999e-50 < l < 7.00000000000000006e134

   1. Initial program 81.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 63.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 7.00000000000000006e134 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 42.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 40.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 56.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-233}:\\ \;\;\;\;d \cdot {\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ t_1 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -1.22 \cdot 10^{-163}:\\ \;\;\;\;t\_1 \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-222}:\\ \;\;\;\;d \cdot {\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-50}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right) \cdot \frac{M\_m \cdot M\_m}{d}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))) (t_1 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -1.22e-163)
     (* t_1 (- 0.0 d))
     (if (<= l 4.1e-222)
       (* d (pow (* t_0 t_0) 0.25))
       (if (<= l 3.5e-50)
         (*
          -0.125
          (* (* (* D_m D_m) (sqrt (/ h (* l (* l l))))) (/ (* M_m M_m) d)))
         (if (<= l 5e+134) (* d t_1) (/ 1.0 (sqrt (* (/ l d) (/ h d))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -1.22e-163) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= 4.1e-222) {
		tmp = d * pow((t_0 * t_0), 0.25);
	} else if (l <= 3.5e-50) {
		tmp = -0.125 * (((D_m * D_m) * sqrt((h / (l * (l * l))))) * ((M_m * M_m) / d));
	} else if (l <= 5e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (l * h)
    t_1 = sqrt(((1.0d0 / h) / l))
    if (l <= (-1.22d-163)) then
        tmp = t_1 * (0.0d0 - d)
    else if (l <= 4.1d-222) then
        tmp = d * ((t_0 * t_0) ** 0.25d0)
    else if (l <= 3.5d-50) then
        tmp = (-0.125d0) * (((d_m * d_m) * sqrt((h / (l * (l * l))))) * ((m_m * m_m) / d))
    else if (l <= 5d+134) then
        tmp = d * t_1
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -1.22e-163) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= 4.1e-222) {
		tmp = d * Math.pow((t_0 * t_0), 0.25);
	} else if (l <= 3.5e-50) {
		tmp = -0.125 * (((D_m * D_m) * Math.sqrt((h / (l * (l * l))))) * ((M_m * M_m) / d));
	} else if (l <= 5e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 / (l * h)
	t_1 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -1.22e-163:
		tmp = t_1 * (0.0 - d)
	elif l <= 4.1e-222:
		tmp = d * math.pow((t_0 * t_0), 0.25)
	elif l <= 3.5e-50:
		tmp = -0.125 * (((D_m * D_m) * math.sqrt((h / (l * (l * l))))) * ((M_m * M_m) / d))
	elif l <= 5e+134:
		tmp = d * t_1
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 / Float64(l * h))
	t_1 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -1.22e-163)
		tmp = Float64(t_1 * Float64(0.0 - d));
	elseif (l <= 4.1e-222)
		tmp = Float64(d * (Float64(t_0 * t_0) ^ 0.25));
	elseif (l <= 3.5e-50)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) * sqrt(Float64(h / Float64(l * Float64(l * l))))) * Float64(Float64(M_m * M_m) / d)));
	elseif (l <= 5e+134)
		tmp = Float64(d * t_1);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 / (l * h);
	t_1 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -1.22e-163)
		tmp = t_1 * (0.0 - d);
	elseif (l <= 4.1e-222)
		tmp = d * ((t_0 * t_0) ^ 0.25);
	elseif (l <= 3.5e-50)
		tmp = -0.125 * (((D_m * D_m) * sqrt((h / (l * (l * l))))) * ((M_m * M_m) / d));
	elseif (l <= 5e+134)
		tmp = d * t_1;
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.22e-163], N[(t$95$1 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.1e-222], N[(d * N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.5e-50], N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+134], N[(d * t$95$1), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.22000000000000003e-163

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.22000000000000003e-163 < l < 4.1000000000000003e-222

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 23.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 23.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{\color{blue}{4}}\right)}\right)\right) \]

      5. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)} \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\left(\frac{1}{4}\right)}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right), \color{blue}{\left(\frac{1}{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{\color{blue}{1}}{4}\right)\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell \cdot h}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{1}{\ell \cdot h}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      15. metadata-eval 45.4%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \frac{1}{4}\right)\right) \]

   9. Applied egg-rr 45.4%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}} \]

   if 4.1000000000000003e-222 < l < 3.49999999999999997e-50

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.0%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{M \cdot D}{d \cdot 4} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)}{\color{blue}{d \cdot 4}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(d \cdot 4\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(\color{blue}{d} \cdot 4\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{h}{\ell}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{-1}{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\frac{\frac{-1}{2}}{\frac{\ell}{h}}\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\ell}{h}\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \left(d \cdot 4\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 78.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\ell, h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{4}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)}{d \cdot 4}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\frac{\ell}{h}}\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell} \cdot h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{d} \cdot \frac{-1}{2}}{\ell}\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{M \cdot D}} \cdot \frac{-1}{2}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{\frac{d}{M \cdot D}}\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{d}{M \cdot D}\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 80.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), h\right)\right), \mathsf{*.f64}\left(d, 4\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 80.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{\frac{-0.5}{\frac{d}{M \cdot D}}}{\ell} \cdot h\right)}}{d \cdot 4}\right)\right) \]

   11. Taylor expanded in d around 0

       \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}}\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2}}{d}}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2}}{d}}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2}}{d}\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}}}{d}\right)\right)\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2}}{d}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2}}{d}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{\color{blue}{2}}}{d}\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{\color{blue}{2}}}{d}\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2}\right), \color{blue}{d}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left(M \cdot M\right), d\right)\right)\right) \]

       18. *-lowering-*.f64 54.9%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), d\right)\right)\right) \]

   13. Simplified 54.9%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right)} \]

   if 3.49999999999999997e-50 < l < 4.99999999999999981e134

   1. Initial program 81.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 63.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 4.99999999999999981e134 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 42.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 40.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 56.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{-222}:\\ \;\;\;\;d \cdot {\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-50}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right) \cdot \frac{M \cdot M}{d}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{d}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot D\_m\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= (* M_m D_m) 1e+30)
   (/ 1.0 (sqrt (* (/ l d) (/ h d))))
   (/
    -1.0
    (* (/ d (* M_m (* D_m (* M_m D_m)))) (* 8.0 (* l (sqrt (/ l h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1e+30) {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	} else {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if ((m_m * d_m) <= 1d+30) then
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    else
        tmp = (-1.0d0) / ((d / (m_m * (d_m * (m_m * d_m)))) * (8.0d0 * (l * sqrt((l / h)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1e+30) {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	} else {
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * Math.sqrt((l / h)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if (M_m * D_m) <= 1e+30:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	else:
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * math.sqrt((l / h)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e+30)
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	else
		tmp = Float64(-1.0 / Float64(Float64(d / Float64(M_m * Float64(D_m * Float64(M_m * D_m)))) * Float64(8.0 * Float64(l * sqrt(Float64(l / h))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if ((M_m * D_m) <= 1e+30)
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	else
		tmp = -1.0 / ((d / (M_m * (D_m * (M_m * D_m)))) * (8.0 * (l * sqrt((l / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+30], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(d / N[(M$95$m * N[(D$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e30

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 32.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 32.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 32.1%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 43.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 43.0%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

   if 1e30 < (*.f64 M D)

   1. Initial program 77.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\sqrt{\ell}\right), \color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right) \cdot \sqrt{d}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \left(\color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right)} \cdot \sqrt{d}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{-1}{2}\right)\right)\right), \color{blue}{\left(\sqrt{d}\right)}\right)\right)\right) \]

   6. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{h}{\ell} \cdot -0.5}{\frac{4 \cdot \left(d \cdot d\right)}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\right)\right) \cdot \sqrt{d}}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(8 \cdot \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot \sqrt{\frac{{\ell}^{3}}{h}}\right) \cdot \color{blue}{8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \frac{d}{{D}^{2} \cdot {M}^{2}}\right) \cdot 8\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt{\frac{{\ell}^{3}}{h}} \cdot \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{{\ell}^{3}}{h}}\right), \color{blue}{\left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{{\ell}^{3}}{h}\right)\right), \left(\color{blue}{\frac{d}{{D}^{2} \cdot {M}^{2}}} \cdot 8\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{3}\right), h\right)\right), \left(\frac{\color{blue}{d}}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot {\ell}^{2}\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \left(\frac{d}{{D}^{2} \cdot {M}^{2}} \cdot 8\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{D}^{2} \cdot {M}^{2}}\right), \color{blue}{8}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{d}{{M}^{2} \cdot {D}^{2}}\right), 8\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{d}{{M}^{2}}}{{D}^{2}}\right), 8\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{{M}^{2}}\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left({M}^{2}\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left({D}^{2}\right)\right), 8\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \left(D \cdot D\right)\right), 8\right)\right)\right) \]

      20. *-lowering-*.f64 33.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right), h\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(D, D\right)\right), 8\right)\right)\right) \]

   9. Simplified 33.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}} \cdot \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot 8\right) \cdot \color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{d}{M \cdot M}}{D \cdot D} \cdot \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{d}{M \cdot M}}{D \cdot D}\right), \color{blue}{\left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       6. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{d}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right), \left(\color{blue}{8} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \left(D \cdot \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \left(M \cdot D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \left(8 \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \color{blue}{\left(\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}\right)}\right)\right)\right) \]

       13. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       16. unpow-prod-down N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\left(\ell \cdot \ell\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{\ell}{h}\right)}^{\frac{1}{2}}}\right)\right)\right)\right) \]

       17. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{\ell \cdot \ell} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       18. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\sqrt{{\ell}^{2}} \cdot {\left(\frac{\color{blue}{\ell}}{h}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       19. sqrt-pow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{\left(\frac{2}{2}\right)} \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left({\ell}^{1} \cdot {\left(\frac{\ell}{\color{blue}{h}}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

       21. unpow1 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \left(\ell \cdot {\color{blue}{\left(\frac{\ell}{h}\right)}}^{\frac{1}{2}}\right)\right)\right)\right) \]

   11. Applied egg-rr 2.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot {\left(\frac{\ell}{h}\right)}^{0.5}\right)\right)}} \]

   12. Taylor expanded in h around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(-1 \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}\right)\right)\right)\right)\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\sqrt{\frac{\ell}{h}}\right)\right)\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\ell}{h}\right)\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 71.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, D\right)\right)\right)\right), \mathsf{*.f64}\left(8, \mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\ell, h\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 71.8%

       \[\leadsto \frac{1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \color{blue}{\left(-\sqrt{\frac{\ell}{h}}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{d}{M \cdot \left(D \cdot \left(M \cdot D\right)\right)} \cdot \left(8 \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ t_1 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{-164}:\\ \;\;\;\;t\_1 \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))) (t_1 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -6e-164)
     (* t_1 (- 0.0 d))
     (if (<= l -5e-310)
       (* d (pow (* t_0 t_0) 0.25))
       (if (<= l 1.5e+134) (* d t_1) (/ 1.0 (sqrt (* (/ l d) (/ h d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -6e-164) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= -5e-310) {
		tmp = d * pow((t_0 * t_0), 0.25);
	} else if (l <= 1.5e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (l * h)
    t_1 = sqrt(((1.0d0 / h) / l))
    if (l <= (-6d-164)) then
        tmp = t_1 * (0.0d0 - d)
    else if (l <= (-5d-310)) then
        tmp = d * ((t_0 * t_0) ** 0.25d0)
    else if (l <= 1.5d+134) then
        tmp = d * t_1
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 / (l * h);
	double t_1 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -6e-164) {
		tmp = t_1 * (0.0 - d);
	} else if (l <= -5e-310) {
		tmp = d * Math.pow((t_0 * t_0), 0.25);
	} else if (l <= 1.5e+134) {
		tmp = d * t_1;
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 / (l * h)
	t_1 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -6e-164:
		tmp = t_1 * (0.0 - d)
	elif l <= -5e-310:
		tmp = d * math.pow((t_0 * t_0), 0.25)
	elif l <= 1.5e+134:
		tmp = d * t_1
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 / Float64(l * h))
	t_1 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -6e-164)
		tmp = Float64(t_1 * Float64(0.0 - d));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(t_0 * t_0) ^ 0.25));
	elseif (l <= 1.5e+134)
		tmp = Float64(d * t_1);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 / (l * h);
	t_1 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -6e-164)
		tmp = t_1 * (0.0 - d);
	elseif (l <= -5e-310)
		tmp = d * ((t_0 * t_0) ^ 0.25);
	elseif (l <= 1.5e+134)
		tmp = d * t_1;
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6e-164], N[(t$95$1 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+134], N[(d * t$95$1), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.0000000000000002e-164

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -6.0000000000000002e-164 < l < -4.999999999999985e-310

   1. Initial program 64.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 12.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 12.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4} + \frac{1}{\color{blue}{4}}\right)}\right)\right) \]

      5. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)} \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(\frac{1}{4}\right)}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(d, \left({\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\left(\frac{1}{4}\right)}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right), \color{blue}{\left(\frac{1}{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{\color{blue}{1}}{4}\right)\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell \cdot h}\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{1}{\ell \cdot h}\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\frac{1}{4}\right)\right)\right) \]

      15. metadata-eval 52.3%

          \[\leadsto \mathsf{*.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \frac{1}{4}\right)\right) \]

   9. Applied egg-rr 52.3%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}} \]

   if -4.999999999999985e-310 < l < 1.49999999999999998e134

   1. Initial program 71.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 45.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 1.49999999999999998e134 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 42.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 40.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 56.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;t\_0 \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-292}:\\ \;\;\;\;d \cdot {\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{-0.25}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -3.3e-165)
     (* t_0 (- 0.0 d))
     (if (<= l -1e-292)
       (* d (pow (* (* l h) (* l h)) -0.25))
       (if (<= l 3.5e+134) (* d t_0) (/ 1.0 (sqrt (* (/ l d) (/ h d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -3.3e-165) {
		tmp = t_0 * (0.0 - d);
	} else if (l <= -1e-292) {
		tmp = d * pow(((l * h) * (l * h)), -0.25);
	} else if (l <= 3.5e+134) {
		tmp = d * t_0;
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (l <= (-3.3d-165)) then
        tmp = t_0 * (0.0d0 - d)
    else if (l <= (-1d-292)) then
        tmp = d * (((l * h) * (l * h)) ** (-0.25d0))
    else if (l <= 3.5d+134) then
        tmp = d * t_0
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -3.3e-165) {
		tmp = t_0 * (0.0 - d);
	} else if (l <= -1e-292) {
		tmp = d * Math.pow(((l * h) * (l * h)), -0.25);
	} else if (l <= 3.5e+134) {
		tmp = d * t_0;
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -3.3e-165:
		tmp = t_0 * (0.0 - d)
	elif l <= -1e-292:
		tmp = d * math.pow(((l * h) * (l * h)), -0.25)
	elif l <= 3.5e+134:
		tmp = d * t_0
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -3.3e-165)
		tmp = Float64(t_0 * Float64(0.0 - d));
	elseif (l <= -1e-292)
		tmp = Float64(d * (Float64(Float64(l * h) * Float64(l * h)) ^ -0.25));
	elseif (l <= 3.5e+134)
		tmp = Float64(d * t_0);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -3.3e-165)
		tmp = t_0 * (0.0 - d);
	elseif (l <= -1e-292)
		tmp = d * (((l * h) * (l * h)) ^ -0.25);
	elseif (l <= 3.5e+134)
		tmp = d * t_0;
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.3e-165], N[(t$95$0 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-292], N[(d * N[Power[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.5e+134], N[(d * t$95$0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.2999999999999998e-165

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 61.6%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -3.2999999999999998e-165 < l < -1.0000000000000001e-292

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 13.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 13.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{d} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{\frac{1}{h}}{\ell}}\right), \color{blue}{d}\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\ell \cdot h}}\right), d\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}}\right), d\right) \]

      5. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right), d\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\ell \cdot h\right)}^{\frac{-1}{2}}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \frac{-1}{2}\right), d\right) \]

      8. *-lowering-*.f64 13.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 13.1%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
   10. Step-by-step derivation

       1. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right), d\right) \]

       2. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right), d\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right), \left(\frac{\frac{-1}{2}}{2}\right)\right), d\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot h\right), \left(\ell \cdot h\right)\right), \left(\frac{\frac{-1}{2}}{2}\right)\right), d\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\ell \cdot h\right)\right), \left(\frac{\frac{-1}{2}}{2}\right)\right), d\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \left(\frac{\frac{-1}{2}}{2}\right)\right), d\right) \]

       7. metadata-eval 50.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \frac{-1}{4}\right), d\right) \]

   11. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{-0.25}} \cdot d \]

   if -1.0000000000000001e-292 < l < 3.50000000000000003e134

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 45.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 3.50000000000000003e134 < l

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 42.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 40.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 56.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-292}:\\ \;\;\;\;d \cdot {\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{-0.25}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -5e-310)
   (* (sqrt (/ (/ 1.0 h) l)) (- 0.0 d))
   (if (<= h 3.2e+19)
     (/ d (sqrt (* l h)))
     (/ 1.0 (sqrt (* (/ l d) (/ h d)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (h <= 3.2e+19) {
		tmp = d / sqrt((l * h));
	} else {
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = sqrt(((1.0d0 / h) / l)) * (0.0d0 - d)
    else if (h <= 3.2d+19) then
        tmp = d / sqrt((l * h))
    else
        tmp = 1.0d0 / sqrt(((l / d) * (h / d)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = Math.sqrt(((1.0 / h) / l)) * (0.0 - d);
	} else if (h <= 3.2e+19) {
		tmp = d / Math.sqrt((l * h));
	} else {
		tmp = 1.0 / Math.sqrt(((l / d) * (h / d)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -5e-310:
		tmp = math.sqrt(((1.0 / h) / l)) * (0.0 - d)
	elif h <= 3.2e+19:
		tmp = d / math.sqrt((l * h))
	else:
		tmp = 1.0 / math.sqrt(((l / d) * (h / d)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(0.0 - d));
	elseif (h <= 3.2e+19)
		tmp = Float64(d / sqrt(Float64(l * h)));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(l / d) * Float64(h / d))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = sqrt(((1.0 / h) / l)) * (0.0 - d);
	elseif (h <= 3.2e+19)
		tmp = d / sqrt((l * h));
	else
		tmp = 1.0 / sqrt(((l / d) * (h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -5e-310], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.2e+19], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(l / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 62.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 52.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 62.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 62.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 41.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 41.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -4.999999999999985e-310 < h < 3.2e19

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 56.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 56.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 56.0%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left(\sqrt{\ell \cdot h}\right)}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

       6. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   11. Applied egg-rr 56.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   if 3.2e19 < h

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 30.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 30.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 29.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 29.4%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{d}{h}} \]

       9. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d}} \cdot \frac{1}{\frac{h}{d}}} \]

       10. frac-times N/A

           \[\leadsto \sqrt{\frac{1 \cdot 1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\frac{1}{\frac{\ell}{d} \cdot \frac{h}{d}}} \]

       12. sqrt-div N/A

           \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}\right)}\right) \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\ell}{d} \cdot \frac{h}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{h}{d}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 42.9%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \mathsf{/.f64}\left(h, d\right)\right)\right)\right) \]

   11. Applied egg-rr 42.9%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d} \cdot \frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq 5.2 \cdot 10^{-228}:\\ \;\;\;\;t\_0 \cdot \left(0 - d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= d 5.2e-228) (* t_0 (- 0.0 d)) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= 5.2e-228) {
		tmp = t_0 * (0.0 - d);
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (d <= 5.2d-228) then
        tmp = t_0 * (0.0d0 - d)
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= 5.2e-228) {
		tmp = t_0 * (0.0 - d);
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if d <= 5.2e-228:
		tmp = t_0 * (0.0 - d)
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (d <= 5.2e-228)
		tmp = Float64(t_0 * Float64(0.0 - d));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (d <= 5.2e-228)
		tmp = t_0 * (0.0 - d);
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 5.2e-228], N[(t$95$0 * N[(0.0 - d), $MachinePrecision]), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 5.2e-228

   1. Initial program 59.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\left(4 \cdot d\right) \cdot d}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d} \cdot \frac{M \cdot D}{d}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{4 \cdot d}\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, \ell\right)}, \frac{-1}{2}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(4 \cdot d\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(d \cdot 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \left(\frac{M \cdot D}{d}\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \color{blue}{\ell}\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 59.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 59.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{d}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{\left(-1 \cdot d\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right), \left(\color{blue}{-1} \cdot d\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right), \left(-1 \cdot d\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 39.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if 5.2e-228 < d

   1. Initial program 74.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 48.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 37.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\frac{h}{d}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -9.2e-256) (sqrt (/ (/ d (/ h d)) l)) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.2e-256) {
		tmp = sqrt(((d / (h / d)) / l));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-9.2d-256)) then
        tmp = sqrt(((d / (h / d)) / l))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9.2e-256) {
		tmp = Math.sqrt(((d / (h / d)) / l));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -9.2e-256:
		tmp = math.sqrt(((d / (h / d)) / l))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -9.2e-256)
		tmp = sqrt(Float64(Float64(d / Float64(h / d)) / l));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -9.2e-256)
		tmp = sqrt(((d / (h / d)) / l));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -9.2e-256], N[Sqrt[N[(N[(d / N[(h / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.199999999999999e-256

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 6.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 6.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 6.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 6.1%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d \cdot \frac{d}{h}}{\ell}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \frac{d}{h}\right), \ell\right)\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \frac{1}{\frac{h}{d}}\right), \ell\right)\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{\frac{h}{d}}\right), \ell\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(\frac{h}{d}\right)\right), \ell\right)\right) \]

       14. /-lowering-/.f64 28.6%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{/.f64}\left(h, d\right)\right), \ell\right)\right) \]

   11. Applied egg-rr 28.6%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\frac{h}{d}}}{\ell}}} \]

   if -9.199999999999999e-256 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 43.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 36.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\frac{h}{d}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -8e-256) (sqrt (/ (/ d (/ h d)) l)) (/ d (sqrt (* l h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -8e-256) {
		tmp = sqrt(((d / (h / d)) / l));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-8d-256)) then
        tmp = sqrt(((d / (h / d)) / l))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -8e-256) {
		tmp = Math.sqrt(((d / (h / d)) / l));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -8e-256:
		tmp = math.sqrt(((d / (h / d)) / l))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -8e-256)
		tmp = sqrt(Float64(Float64(d / Float64(h / d)) / l));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -8e-256)
		tmp = sqrt(((d / (h / d)) / l));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -8e-256], N[Sqrt[N[(N[(d / N[(h / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.99999999999999982e-256

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 6.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 6.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 6.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 6.1%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       5. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       6. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d \cdot \frac{d}{h}}{\ell}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \frac{d}{h}\right), \ell\right)\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(d \cdot \frac{1}{\frac{h}{d}}\right), \ell\right)\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{d}{\frac{h}{d}}\right), \ell\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(\frac{h}{d}\right)\right), \ell\right)\right) \]

       14. /-lowering-/.f64 28.6%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{/.f64}\left(h, d\right)\right), \ell\right)\right) \]

   11. Applied egg-rr 28.6%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\frac{h}{d}}}{\ell}}} \]

   if -7.99999999999999982e-256 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

      5. /-lowering-/.f64 43.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

   9. Applied egg-rr 42.9%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       2. metadata-eval N/A

          \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       3. un-div-inv N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left(\sqrt{\ell \cdot h}\right)}\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

       6. *-lowering-*.f64 43.0%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   11. Applied egg-rr 43.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 26.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. associate-*l* N/A

      \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   10. unpow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

3. Simplified 52.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right) \]

   5. /-lowering-/.f64 26.5%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right) \]

7. Simplified 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
8. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right)\right) \]

   4. *-lowering-*.f64 26.2%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right)\right) \]

9. Applied egg-rr 26.2%

   \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]
10. Step-by-step derivation

    1. sqrt-div N/A

       \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\ell \cdot h}}} \]

    2. metadata-eval N/A

       \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

    3. un-div-inv N/A

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left(\sqrt{\ell \cdot h}\right)}\right) \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

    6. *-lowering-*.f64 26.2%

       \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

11. Applied egg-rr 26.2%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))