Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 78.5%

Time: 25.8s

Alternatives: 23

Speedup: 1.4×

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: 65.7% 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: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\frac{D\_m}{\frac{d \cdot 4}{M\_m}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{-286}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{\sqrt{0 - \ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_0\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{h} \cdot \left(D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot {\left(\frac{h}{\ell}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0
         (+
          1.0
          (/
           (* (/ D_m (/ d M_m)) (* (/ D_m (/ (* d 4.0) M_m)) (* h -0.5)))
           l))))
   (if (<= d -1.55e-286)
     (* (/ (pow (- 0.0 d) 0.5) (sqrt (- 0.0 l))) (* (sqrt (/ d h)) t_0))
     (if (<= d 5.6e-226)
       (*
        (/ 1.0 h)
        (+
         (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
         (* d (pow (/ h l) 0.5))))
       (* (sqrt (/ d l)) (* t_0 (/ (sqrt d) (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 = 1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l);
	double tmp;
	if (d <= -1.55e-286) {
		tmp = (pow((0.0 - d), 0.5) / sqrt((0.0 - l))) * (sqrt((d / h)) * t_0);
	} else if (d <= 5.6e-226) {
		tmp = (1.0 / h) * ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * pow((h / l), 0.5)));
	} else {
		tmp = sqrt((d / l)) * (t_0 * (sqrt(d) / 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 = 1.0d0 + (((d_m / (d / m_m)) * ((d_m / ((d * 4.0d0) / m_m)) * (h * (-0.5d0)))) / l)
    if (d <= (-1.55d-286)) then
        tmp = (((0.0d0 - d) ** 0.5d0) / sqrt((0.0d0 - l))) * (sqrt((d / h)) * t_0)
    else if (d <= 5.6d-226) then
        tmp = (1.0d0 / h) * ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * ((h / l) ** 0.5d0)))
    else
        tmp = sqrt((d / l)) * (t_0 * (sqrt(d) / 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 = 1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l);
	double tmp;
	if (d <= -1.55e-286) {
		tmp = (Math.pow((0.0 - d), 0.5) / Math.sqrt((0.0 - l))) * (Math.sqrt((d / h)) * t_0);
	} else if (d <= 5.6e-226) {
		tmp = (1.0 / h) * ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.pow((h / l), 0.5)));
	} else {
		tmp = Math.sqrt((d / l)) * (t_0 * (Math.sqrt(d) / 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 = 1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l)
	tmp = 0
	if d <= -1.55e-286:
		tmp = (math.pow((0.0 - d), 0.5) / math.sqrt((0.0 - l))) * (math.sqrt((d / h)) * t_0)
	elif d <= 5.6e-226:
		tmp = (1.0 / h) * ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.pow((h / l), 0.5)))
	else:
		tmp = math.sqrt((d / l)) * (t_0 * (math.sqrt(d) / 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(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m / Float64(Float64(d * 4.0) / M_m)) * Float64(h * -0.5))) / l))
	tmp = 0.0
	if (d <= -1.55e-286)
		tmp = Float64(Float64((Float64(0.0 - d) ^ 0.5) / sqrt(Float64(0.0 - l))) * Float64(sqrt(Float64(d / h)) * t_0));
	elseif (d <= 5.6e-226)
		tmp = Float64(Float64(1.0 / h) * Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * (Float64(h / l) ^ 0.5))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_0 * Float64(sqrt(d) / 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 = 1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l);
	tmp = 0.0;
	if (d <= -1.55e-286)
		tmp = (((0.0 - d) ^ 0.5) / sqrt((0.0 - l))) * (sqrt((d / h)) * t_0);
	elseif (d <= 5.6e-226)
		tmp = (1.0 / h) * ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * ((h / l) ^ 0.5)));
	else
		tmp = sqrt((d / l)) * (t_0 * (sqrt(d) / 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[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / N[(N[(d * 4.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.55e-286], 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] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.6e-226], N[(N[(1.0 / h), $MachinePrecision] * N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.54999999999999991e-286

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 57.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. 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 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 69.8%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. frac-2neg N/A

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

       2. sqrt-div N/A

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

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

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

       4. pow1/2 N/A

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

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

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

       6. sub0-neg N/A

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

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

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

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

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

       9. neg-sub0 N/A

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

       10. --lowering--.f64 78.7%

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

   12. Applied egg-rr 78.7%

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

   if -1.54999999999999991e-286 < d < 5.60000000000000016e-226

   1. Initial program 36.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 9.3%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 22.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\frac{1}{h} \cdot \left(D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot {\left(\frac{h}{\ell}\right)}^{0.5}\right)} \]

   if 5.60000000000000016e-226 < d

   1. Initial program 71.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 57.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 74.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.0%

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

   10. Applied egg-rr 76.0%

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

       1. sqrt-div N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 83.8%

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

   12. Applied egg-rr 83.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-286}:\\ \;\;\;\;\frac{{\left(0 - d\right)}^{0.5}}{\sqrt{0 - \ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{h} \cdot \left(D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot {\left(\frac{h}{\ell}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d \cdot 4}{M\_m}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\frac{t\_0}{D\_m}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\frac{D\_m}{t\_0} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* d 4.0) M_m)))
   (if (<= d -3.2e-211)
     (*
      (/ (sqrt (- 0.0 d)) (pow (- 0.0 l) 0.5))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* h -0.5) (/ (* D_m (/ M_m d)) (/ t_0 D_m))) l))))
     (if (<= d 4.2e-224)
       (/
        (+
         (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
         (* d (sqrt (/ h l))))
        h)
       (*
        (sqrt (/ d l))
        (*
         (+ 1.0 (/ (* (/ D_m (/ d M_m)) (* (/ D_m t_0) (* h -0.5))) l))
         (/ (sqrt d) (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 * 4.0) / M_m;
	double tmp;
	if (d <= -3.2e-211) {
		tmp = (sqrt((0.0 - d)) / pow((0.0 - l), 0.5)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M_m / d)) / (t_0 / D_m))) / l)));
	} else if (d <= 4.2e-224) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else {
		tmp = sqrt((d / l)) * ((1.0 + (((D_m / (d / M_m)) * ((D_m / t_0) * (h * -0.5))) / l)) * (sqrt(d) / 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 * 4.0d0) / m_m
    if (d <= (-3.2d-211)) then
        tmp = (sqrt((0.0d0 - d)) / ((0.0d0 - l) ** 0.5d0)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m_m / d)) / (t_0 / d_m))) / l)))
    else if (d <= 4.2d-224) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else
        tmp = sqrt((d / l)) * ((1.0d0 + (((d_m / (d / m_m)) * ((d_m / t_0) * (h * (-0.5d0)))) / l)) * (sqrt(d) / 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 * 4.0) / M_m;
	double tmp;
	if (d <= -3.2e-211) {
		tmp = (Math.sqrt((0.0 - d)) / Math.pow((0.0 - l), 0.5)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M_m / d)) / (t_0 / D_m))) / l)));
	} else if (d <= 4.2e-224) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else {
		tmp = Math.sqrt((d / l)) * ((1.0 + (((D_m / (d / M_m)) * ((D_m / t_0) * (h * -0.5))) / l)) * (Math.sqrt(d) / 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 * 4.0) / M_m
	tmp = 0
	if d <= -3.2e-211:
		tmp = (math.sqrt((0.0 - d)) / math.pow((0.0 - l), 0.5)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M_m / d)) / (t_0 / D_m))) / l)))
	elif d <= 4.2e-224:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	else:
		tmp = math.sqrt((d / l)) * ((1.0 + (((D_m / (d / M_m)) * ((D_m / t_0) * (h * -0.5))) / l)) * (math.sqrt(d) / 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(Float64(d * 4.0) / M_m)
	tmp = 0.0
	if (d <= -3.2e-211)
		tmp = Float64(Float64(sqrt(Float64(0.0 - d)) / (Float64(0.0 - l) ^ 0.5)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * Float64(Float64(D_m * Float64(M_m / d)) / Float64(t_0 / D_m))) / l))));
	elseif (d <= 4.2e-224)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	else
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m / t_0) * Float64(h * -0.5))) / l)) * Float64(sqrt(d) / 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 * 4.0) / M_m;
	tmp = 0.0;
	if (d <= -3.2e-211)
		tmp = (sqrt((0.0 - d)) / ((0.0 - l) ^ 0.5)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M_m / d)) / (t_0 / D_m))) / l)));
	elseif (d <= 4.2e-224)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	else
		tmp = sqrt((d / l)) * ((1.0 + (((D_m / (d / M_m)) * ((D_m / t_0) * (h * -0.5))) / l)) * (sqrt(d) / 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[(N[(d * 4.0), $MachinePrecision] / M$95$m), $MachinePrecision]}, If[LessEqual[d, -3.2e-211], N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $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[(h * -0.5), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-224], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / t$95$0), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.19999999999999985e-211

   1. Initial program 74.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 62.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 75.2%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

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

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

      5. pow1/2 N/A

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

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

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

      7. neg-sub0 N/A

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

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

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

      9. pow1/2 N/A

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

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

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 82.6%

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

   10. Applied egg-rr 82.6%

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

   if -3.19999999999999985e-211 < d < 4.20000000000000013e-224

   1. Initial program 31.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 13.8%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 17.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 57.6%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 4.20000000000000013e-224 < d

   1. Initial program 71.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 57.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 74.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.0%

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

   10. Applied egg-rr 76.0%

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

       1. sqrt-div N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 83.8%

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

   12. Applied egg-rr 83.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot \frac{D \cdot \frac{M}{d}}{\frac{\frac{d \cdot 4}{M}}{D}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-124}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{\left(h \cdot -0.5\right) \cdot \frac{D\_m \cdot M\_m}{d \cdot 4}}{\frac{d}{D\_m \cdot M\_m}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\frac{D\_m}{\frac{d \cdot 4}{M\_m}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -1.3e-124)
     (*
      t_0
      (*
       (sqrt (/ d h))
       (+
        1.0
        (/ (/ (* (* h -0.5) (/ (* D_m M_m) (* d 4.0))) (/ d (* D_m M_m))) l))))
     (if (<= d 1.1e-225)
       (/
        (+
         (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
         (* d (sqrt (/ h l))))
        h)
       (*
        t_0
        (*
         (+
          1.0
          (/ (* (/ D_m (/ d M_m)) (* (/ D_m (/ (* d 4.0) M_m)) (* h -0.5))) l))
         (/ (sqrt d) (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 <= -1.3e-124) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 + ((((h * -0.5) * ((D_m * M_m) / (d * 4.0))) / (d / (D_m * M_m))) / l)));
	} else if (d <= 1.1e-225) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else {
		tmp = t_0 * ((1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l)) * (sqrt(d) / 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 <= (-1.3d-124)) then
        tmp = t_0 * (sqrt((d / h)) * (1.0d0 + ((((h * (-0.5d0)) * ((d_m * m_m) / (d * 4.0d0))) / (d / (d_m * m_m))) / l)))
    else if (d <= 1.1d-225) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else
        tmp = t_0 * ((1.0d0 + (((d_m / (d / m_m)) * ((d_m / ((d * 4.0d0) / m_m)) * (h * (-0.5d0)))) / l)) * (sqrt(d) / 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 <= -1.3e-124) {
		tmp = t_0 * (Math.sqrt((d / h)) * (1.0 + ((((h * -0.5) * ((D_m * M_m) / (d * 4.0))) / (d / (D_m * M_m))) / l)));
	} else if (d <= 1.1e-225) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else {
		tmp = t_0 * ((1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l)) * (Math.sqrt(d) / 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 <= -1.3e-124:
		tmp = t_0 * (math.sqrt((d / h)) * (1.0 + ((((h * -0.5) * ((D_m * M_m) / (d * 4.0))) / (d / (D_m * M_m))) / l)))
	elif d <= 1.1e-225:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	else:
		tmp = t_0 * ((1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l)) * (math.sqrt(d) / 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 <= -1.3e-124)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(Float64(h * -0.5) * Float64(Float64(D_m * M_m) / Float64(d * 4.0))) / Float64(d / Float64(D_m * M_m))) / l))));
	elseif (d <= 1.1e-225)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m / Float64(Float64(d * 4.0) / M_m)) * Float64(h * -0.5))) / l)) * Float64(sqrt(d) / 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 <= -1.3e-124)
		tmp = t_0 * (sqrt((d / h)) * (1.0 + ((((h * -0.5) * ((D_m * M_m) / (d * 4.0))) / (d / (D_m * M_m))) / l)));
	elseif (d <= 1.1e-225)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	else
		tmp = t_0 * ((1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l)) * (sqrt(d) / 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, -1.3e-124], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(h * -0.5), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-225], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / N[(N[(d * 4.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.3e-124

   1. Initial program 79.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 80.0%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 80.1%

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

   10. Applied egg-rr 80.1%

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

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

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

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

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

       7. associate-/r/ N/A

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

       8. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

       13. associate-/l/ N/A

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

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

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

       15. *-lowering-*.f64 81.2%

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

   12. Applied egg-rr 81.2%

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

   if -1.3e-124 < d < 1.1e-225

   1. Initial program 36.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 18.9%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 19.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 54.8%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.1e-225 < d

   1. Initial program 71.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 57.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 74.5%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.0%

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

   10. Applied egg-rr 76.0%

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

       1. sqrt-div N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 83.8%

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

   12. Applied egg-rr 83.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{\left(h \cdot -0.5\right) \cdot \frac{D \cdot M}{d \cdot 4}}{\frac{d}{D \cdot M}}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-125}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + \left(\frac{D\_m \cdot M\_m}{d \cdot 4} \cdot \frac{D\_m \cdot M\_m}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;d \leq 7.3 \cdot 10^{+78}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \left(M\_m \cdot \frac{h}{d}\right)\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d h))))
   (if (<= d -9.5e-125)
     (*
      t_0
      (*
       t_1
       (+
        1.0
        (* (* (/ (* D_m M_m) (* d 4.0)) (/ (* D_m M_m) d)) (* -0.5 (/ h l))))))
     (if (<= d 1.2e-222)
       (/
        (+
         (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
         (* d (sqrt (/ h l))))
        h)
       (if (<= d 7.3e+78)
         (*
          t_0
          (*
           t_1
           (+
            1.0
            (/ (* (/ D_m (/ d M_m)) (* (* D_m -0.125) (* M_m (/ h d)))) l))))
         (*
          (/ d (pow (* l h) 0.5))
          (-
           1.0
           (/
            (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 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 t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double tmp;
	if (d <= -9.5e-125) {
		tmp = t_0 * (t_1 * (1.0 + ((((D_m * M_m) / (d * 4.0)) * ((D_m * M_m) / d)) * (-0.5 * (h / l)))));
	} else if (d <= 1.2e-222) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else if (d <= 7.3e+78) {
		tmp = t_0 * (t_1 * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	} else {
		tmp = (d / pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    if (d <= (-9.5d-125)) then
        tmp = t_0 * (t_1 * (1.0d0 + ((((d_m * m_m) / (d * 4.0d0)) * ((d_m * m_m) / d)) * ((-0.5d0) * (h / l)))))
    else if (d <= 1.2d-222) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else if (d <= 7.3d+78) then
        tmp = t_0 * (t_1 * (1.0d0 + (((d_m / (d / m_m)) * ((d_m * (-0.125d0)) * (m_m * (h / d)))) / l)))
    else
        tmp = (d / ((l * h) ** 0.5d0)) * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * 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 t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (d <= -9.5e-125) {
		tmp = t_0 * (t_1 * (1.0 + ((((D_m * M_m) / (d * 4.0)) * ((D_m * M_m) / d)) * (-0.5 * (h / l)))));
	} else if (d <= 1.2e-222) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else if (d <= 7.3e+78) {
		tmp = t_0 * (t_1 * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	} else {
		tmp = (d / Math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if d <= -9.5e-125:
		tmp = t_0 * (t_1 * (1.0 + ((((D_m * M_m) / (d * 4.0)) * ((D_m * M_m) / d)) * (-0.5 * (h / l)))))
	elif d <= 1.2e-222:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	elif d <= 7.3e+78:
		tmp = t_0 * (t_1 * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)))
	else:
		tmp = (d / math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -9.5e-125)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(Float64(Float64(Float64(D_m * M_m) / Float64(d * 4.0)) * Float64(Float64(D_m * M_m) / d)) * Float64(-0.5 * Float64(h / l))))));
	elseif (d <= 1.2e-222)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (d <= 7.3e+78)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m * -0.125) * Float64(M_m * Float64(h / d)))) / l))));
	else
		tmp = Float64(Float64(d / (Float64(l * h) ^ 0.5)) * 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)));
	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 = sqrt((d / h));
	tmp = 0.0;
	if (d <= -9.5e-125)
		tmp = t_0 * (t_1 * (1.0 + ((((D_m * M_m) / (d * 4.0)) * ((D_m * M_m) / d)) * (-0.5 * (h / l)))));
	elseif (d <= 1.2e-222)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	elseif (d <= 7.3e+78)
		tmp = t_0 * (t_1 * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	else
		tmp = (d / ((l * h) ^ 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -9.5e-125], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e-222], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7.3e+78], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(M$95$m * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $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]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.50000000000000031e-125

   1. Initial program 79.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 79.6%

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

   6. Applied egg-rr 79.6%

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

   if -9.50000000000000031e-125 < d < 1.19999999999999997e-222

   1. Initial program 37.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 18.5%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 19.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 55.6%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 1.19999999999999997e-222 < d < 7.3e78

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 67.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 75.3%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 76.6%

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

   10. Applied egg-rr 76.6%

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

   11. Taylor expanded in D around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. associate-/l* N/A

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

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

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

       7. /-lowering-/.f64 73.1%

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

   13. Simplified 73.1%

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

   if 7.3e78 < d

   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. Add Preprocessing

   4. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(\frac{D \cdot M}{d \cdot 4} \cdot \frac{D \cdot M}{d}\right) \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;d \leq 7.3 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \frac{h}{d}\right)\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+132}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\frac{D\_m}{\frac{d \cdot 4}{M\_m}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -3.05e+103)
   (/
    (+
     (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
     (* d (sqrt (/ h l))))
    h)
   (if (<= l 6.6e+132)
     (*
      (*
       (sqrt (/ d h))
       (+
        1.0
        (/ (* (/ D_m (/ d M_m)) (* (/ D_m (/ (* d 4.0) M_m)) (* h -0.5))) l)))
      (sqrt (/ d l)))
     (* d (/ (pow h -0.5) (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 <= -3.05e+103) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else if (l <= 6.6e+132) {
		tmp = (sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l))) * sqrt((d / l));
	} else {
		tmp = d * (pow(h, -0.5) / 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 <= (-3.05d+103)) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else if (l <= 6.6d+132) then
        tmp = (sqrt((d / h)) * (1.0d0 + (((d_m / (d / m_m)) * ((d_m / ((d * 4.0d0) / m_m)) * (h * (-0.5d0)))) / l))) * sqrt((d / l))
    else
        tmp = d * ((h ** (-0.5d0)) / 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 <= -3.05e+103) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else if (l <= 6.6e+132) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l))) * Math.sqrt((d / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / 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 <= -3.05e+103:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	elif l <= 6.6e+132:
		tmp = (math.sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l))) * math.sqrt((d / l))
	else:
		tmp = d * (math.pow(h, -0.5) / 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 <= -3.05e+103)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (l <= 6.6e+132)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m / Float64(Float64(d * 4.0) / M_m)) * Float64(h * -0.5))) / l))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / 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 <= -3.05e+103)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	elseif (l <= 6.6e+132)
		tmp = (sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m / ((d * 4.0) / M_m)) * (h * -0.5))) / l))) * sqrt((d / l));
	else
		tmp = d * ((h ^ -0.5) / 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, -3.05e+103], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[l, 6.6e+132], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / N[(N[(d * 4.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.0500000000000001e103

   1. Initial program 46.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 32.5%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 13.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 49.9%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if -3.0500000000000001e103 < l < 6.6000000000000006e132

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 62.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 79.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 79.4%

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

   10. Applied egg-rr 79.4%

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

   if 6.6000000000000006e132 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 48.6%

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

   7. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

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

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+132}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\frac{D}{\frac{d \cdot 4}{M}} \cdot \left(h \cdot -0.5\right)\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}} \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \left(M\_m \cdot \frac{h}{d}\right)\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -6.6e+72)
   (/
    (+
     (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
     (* d (sqrt (/ h l))))
    h)
   (if (<= l 2.05e+130)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (/ D_m (/ d M_m)) (* (* D_m -0.125) (* M_m (/ h d)))) l))))
     (* d (/ (pow h -0.5) (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 <= -6.6e+72) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else if (l <= 2.05e+130) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	} else {
		tmp = d * (pow(h, -0.5) / 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 <= (-6.6d+72)) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else if (l <= 2.05d+130) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((d_m / (d / m_m)) * ((d_m * (-0.125d0)) * (m_m * (h / d)))) / l)))
    else
        tmp = d * ((h ** (-0.5d0)) / 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 <= -6.6e+72) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else if (l <= 2.05e+130) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / 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 <= -6.6e+72:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	elif l <= 2.05e+130:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)))
	else:
		tmp = d * (math.pow(h, -0.5) / 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 <= -6.6e+72)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (l <= 2.05e+130)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) * Float64(Float64(D_m * -0.125) * Float64(M_m * Float64(h / d)))) / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / 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 <= -6.6e+72)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	elseif (l <= 2.05e+130)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((D_m / (d / M_m)) * ((D_m * -0.125) * (M_m * (h / d)))) / l)));
	else
		tmp = d * ((h ^ -0.5) / 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, -6.6e+72], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[l, 2.05e+130], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(M$95$m * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.6e72

   1. Initial program 52.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.3%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 20.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 56.5%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if -6.6e72 < l < 2.04999999999999989e130

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. 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 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 79.0%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 79.3%

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

   10. Applied egg-rr 79.3%

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

   11. Taylor expanded in D around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. associate-/l* N/A

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

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

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

       7. /-lowering-/.f64 75.7%

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

   13. Simplified 75.7%

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

   if 2.04999999999999989e130 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 48.6%

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

   7. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

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

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}} \cdot \left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \frac{h}{d}\right)\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{D\_m \cdot \left(\frac{\left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+131}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -2.45e-293)
   (/
    (+
     (* D_m (* (/ (* (* M_m (* D_m M_m)) -0.125) d) (pow (/ h l) 1.5)))
     (* d (sqrt (/ h l))))
    h)
   (if (<= l 2.25e+131)
     (*
      (/ d (pow (* l h) 0.5))
      (-
       1.0
       (/ (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h)) l)))
     (* d (/ (pow h -0.5) (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 <= -2.45e-293) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * pow((h / l), 1.5))) + (d * sqrt((h / l)))) / h;
	} else if (l <= 2.25e+131) {
		tmp = (d / pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = d * (pow(h, -0.5) / 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 <= (-2.45d-293)) then
        tmp = ((d_m * ((((m_m * (d_m * m_m)) * (-0.125d0)) / d) * ((h / l) ** 1.5d0))) + (d * sqrt((h / l)))) / h
    else if (l <= 2.25d+131) then
        tmp = (d / ((l * h) ** 0.5d0)) * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / 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 <= -2.45e-293) {
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * Math.pow((h / l), 1.5))) + (d * Math.sqrt((h / l)))) / h;
	} else if (l <= 2.25e+131) {
		tmp = (d / Math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / 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 <= -2.45e-293:
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * math.pow((h / l), 1.5))) + (d * math.sqrt((h / l)))) / h
	elif l <= 2.25e+131:
		tmp = (d / math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / 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 <= -2.45e-293)
		tmp = Float64(Float64(Float64(D_m * Float64(Float64(Float64(Float64(M_m * Float64(D_m * M_m)) * -0.125) / d) * (Float64(h / l) ^ 1.5))) + Float64(d * sqrt(Float64(h / l)))) / h);
	elseif (l <= 2.25e+131)
		tmp = Float64(Float64(d / (Float64(l * h) ^ 0.5)) * 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)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / 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 <= -2.45e-293)
		tmp = ((D_m * ((((M_m * (D_m * M_m)) * -0.125) / d) * ((h / l) ^ 1.5))) + (d * sqrt((h / l)))) / h;
	elseif (l <= 2.25e+131)
		tmp = (d / ((l * h) ^ 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	else
		tmp = d * ((h ^ -0.5) / 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, -2.45e-293], N[(N[(N[(D$95$m * N[(N[(N[(N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[l, 2.25e+131], N[(N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $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], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-293

   1. Initial program 67.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. Taylor expanded in h around 0

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

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

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

   7. Simplified 24.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. times-frac N/A

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

      14. sqrt-prod N/A

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

      15. times-frac N/A

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

   9. Applied egg-rr 61.0%

      \[\leadsto \frac{\color{blue}{D \cdot \left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right)} + d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if -2.45e-293 < l < 2.2500000000000001e131

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

   if 2.2500000000000001e131 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 48.6%

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

   7. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

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

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot -0.125}{d} \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+131}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.38 \cdot 10^{-204}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.38e-204)
   (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -2.45e-293)
     (/ d (pow (* (* l h) (* l h)) 0.25))
     (if (<= l 3.7e+132)
       (*
        (/ d (pow (* l h) 0.5))
        (-
         1.0
         (/ (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h)) l)))
       (* d (/ (pow h -0.5) (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 <= -1.38e-204) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (l <= -2.45e-293) {
		tmp = d / pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 3.7e+132) {
		tmp = (d / pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = d * (pow(h, -0.5) / 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 <= (-1.38d-204)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (l <= (-2.45d-293)) then
        tmp = d / (((l * h) * (l * h)) ** 0.25d0)
    else if (l <= 3.7d+132) then
        tmp = (d / ((l * h) ** 0.5d0)) * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / 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 <= -1.38e-204) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -2.45e-293) {
		tmp = d / Math.pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 3.7e+132) {
		tmp = (d / Math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / 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 <= -1.38e-204:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif l <= -2.45e-293:
		tmp = d / math.pow(((l * h) * (l * h)), 0.25)
	elif l <= 3.7e+132:
		tmp = (d / math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / 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 <= -1.38e-204)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -2.45e-293)
		tmp = Float64(d / (Float64(Float64(l * h) * Float64(l * h)) ^ 0.25));
	elseif (l <= 3.7e+132)
		tmp = Float64(Float64(d / (Float64(l * h) ^ 0.5)) * 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)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / 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 <= -1.38e-204)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (l <= -2.45e-293)
		tmp = d / (((l * h) * (l * h)) ^ 0.25);
	elseif (l <= 3.7e+132)
		tmp = (d / ((l * h) ^ 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	else
		tmp = d * ((h ^ -0.5) / 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, -1.38e-204], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.45e-293], N[(d / N[Power[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e+132], N[(N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $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], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.3799999999999999e-204

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. 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 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 43.1%

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

   if -1.3799999999999999e-204 < l < -2.45e-293

   1. Initial program 94.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 43.8%

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

   7. Simplified 43.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 43.8%

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

   9. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\ell \cdot h\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{4}}}\right)\right) \]

       4. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right), \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \frac{1}{4}\right)\right) \]

   11. Applied egg-rr 63.9%

       \[\leadsto \frac{d}{\color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}} \]

   if -2.45e-293 < l < 3.70000000000000011e132

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

   if 3.70000000000000011e132 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 48.6%

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

   7. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

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

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.38 \cdot 10^{-204}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-206}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -1.02e-206)
   (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -2.45e-293)
     (/ d (pow (* (* l h) (* l h)) 0.25))
     (if (<= l 8.8e+124)
       (*
        (/ d (pow (* l h) 0.5))
        (-
         1.0
         (/ (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 h)) l)))
       (/ (/ d (sqrt 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 tmp;
	if (l <= -1.02e-206) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (l <= -2.45e-293) {
		tmp = d / pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 8.8e+124) {
		tmp = (d / pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = (d / sqrt(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) :: tmp
    if (l <= (-1.02d-206)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (l <= (-2.45d-293)) then
        tmp = d / (((l * h) * (l * h)) ** 0.25d0)
    else if (l <= 8.8d+124) then
        tmp = (d / ((l * h) ** 0.5d0)) * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * h)) / l))
    else
        tmp = (d / sqrt(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 tmp;
	if (l <= -1.02e-206) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -2.45e-293) {
		tmp = d / Math.pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 8.8e+124) {
		tmp = (d / Math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	} else {
		tmp = (d / Math.sqrt(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):
	tmp = 0
	if l <= -1.02e-206:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif l <= -2.45e-293:
		tmp = d / math.pow(((l * h) * (l * h)), 0.25)
	elif l <= 8.8e+124:
		tmp = (d / math.pow((l * h), 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l))
	else:
		tmp = (d / math.sqrt(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)
	tmp = 0.0
	if (l <= -1.02e-206)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -2.45e-293)
		tmp = Float64(d / (Float64(Float64(l * h) * Float64(l * h)) ^ 0.25));
	elseif (l <= 8.8e+124)
		tmp = Float64(Float64(d / (Float64(l * h) ^ 0.5)) * 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)));
	else
		tmp = Float64(Float64(d / sqrt(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)
	tmp = 0.0;
	if (l <= -1.02e-206)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (l <= -2.45e-293)
		tmp = d / (((l * h) * (l * h)) ^ 0.25);
	elseif (l <= 8.8e+124)
		tmp = (d / ((l * h) ^ 0.5)) * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * h)) / l));
	else
		tmp = (d / sqrt(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_] := If[LessEqual[l, -1.02e-206], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.45e-293], N[(d / N[Power[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.8e+124], N[(N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $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], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.0200000000000001e-206

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. 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 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 43.1%

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

   if -1.0200000000000001e-206 < l < -2.45e-293

   1. Initial program 94.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 43.8%

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

   7. Simplified 43.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 43.8%

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

   9. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\ell \cdot h\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{4}}}\right)\right) \]

       4. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right), \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \frac{1}{4}\right)\right) \]

   11. Applied egg-rr 63.9%

       \[\leadsto \frac{d}{\color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}} \]

   if -2.45e-293 < l < 8.8000000000000004e124

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]

   if 8.8000000000000004e124 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 48.6%

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

   7. Simplified 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 48.6%

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

   9. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{d}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \left(\sqrt{\ell}\right)\right), \left(\sqrt{\color{blue}{h}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\ell\right)\right), \left(\sqrt{h}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 74.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\ell\right)\right), \mathsf{sqrt.f64}\left(h\right)\right) \]

   11. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-206}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{-43}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-205}:\\ \;\;\;\;h \cdot \left(\sqrt{\frac{1}{\ell \cdot \left(h \cdot \left(\ell \cdot \ell\right)\right)}} \cdot \left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+138}:\\ \;\;\;\;t\_0 \cdot \left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}}}{\frac{d \cdot 4}{D\_m \cdot M\_m}} \cdot \frac{h \cdot -0.5}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 (pow (* l h) 0.5))))
   (if (<= d -1.4e-43)
     (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= d 1.55e-205)
       (*
        h
        (*
         (sqrt (/ 1.0 (* l (* h (* l l)))))
         (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 d))))
       (if (<= d 5.5e+138)
         (*
          t_0
          (+
           1.0
           (*
            (/ (/ D_m (/ d M_m)) (/ (* d 4.0) (* D_m M_m)))
            (/ (* h -0.5) l))))
         (*
          t_0
          (-
           1.0
           (/
            (* (* (* (* M_m M_m) 0.25) (* (/ D_m d) (/ D_m d))) (* 0.5 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 t_0 = d / pow((l * h), 0.5);
	double tmp;
	if (d <= -1.4e-43) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (d <= 1.55e-205) {
		tmp = h * (sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)));
	} else if (d <= 5.5e+138) {
		tmp = t_0 * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	} else {
		tmp = t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = d / ((l * h) ** 0.5d0)
    if (d <= (-1.4d-43)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (d <= 1.55d-205) then
        tmp = h * (sqrt((1.0d0 / (l * (h * (l * l))))) * ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / d)))
    else if (d <= 5.5d+138) then
        tmp = t_0 * (1.0d0 + (((d_m / (d / m_m)) / ((d * 4.0d0) / (d_m * m_m))) * ((h * (-0.5d0)) / l)))
    else
        tmp = t_0 * (1.0d0 - (((((m_m * m_m) * 0.25d0) * ((d_m / d) * (d_m / d))) * (0.5d0 * 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 t_0 = d / Math.pow((l * h), 0.5);
	double tmp;
	if (d <= -1.4e-43) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (d <= 1.55e-205) {
		tmp = h * (Math.sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)));
	} else if (d <= 5.5e+138) {
		tmp = t_0 * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	} else {
		tmp = t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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):
	t_0 = d / math.pow((l * h), 0.5)
	tmp = 0
	if d <= -1.4e-43:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif d <= 1.55e-205:
		tmp = h * (math.sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)))
	elif d <= 5.5e+138:
		tmp = t_0 * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)))
	else:
		tmp = t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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)
	t_0 = Float64(d / (Float64(l * h) ^ 0.5))
	tmp = 0.0
	if (d <= -1.4e-43)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= 1.55e-205)
		tmp = Float64(h * Float64(sqrt(Float64(1.0 / Float64(l * Float64(h * Float64(l * l))))) * Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / d))));
	elseif (d <= 5.5e+138)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) / Float64(Float64(d * 4.0) / Float64(D_m * M_m))) * Float64(Float64(h * -0.5) / l))));
	else
		tmp = 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)));
	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 / ((l * h) ^ 0.5);
	tmp = 0.0;
	if (d <= -1.4e-43)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (d <= 1.55e-205)
		tmp = h * (sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)));
	elseif (d <= 5.5e+138)
		tmp = t_0 * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	else
		tmp = t_0 * (1.0 - (((((M_m * M_m) * 0.25) * ((D_m / d) * (D_m / d))) * (0.5 * 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_] := Block[{t$95$0 = N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.4e-43], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-205], N[(h * N[(N[Sqrt[N[(1.0 / N[(l * N[(h * N[(l * l), $MachinePrecision]), $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]), $MachinePrecision], If[LessEqual[d, 5.5e+138], N[(t$95$0 * N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3999999999999999e-43

   1. Initial program 80.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 67.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 51.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 51.8%

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

   if -1.3999999999999999e-43 < d < 1.54999999999999991e-205

   1. Initial program 44.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 30.2%

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

   5. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)} \]

   7. Simplified 33.4%

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

   8. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

   10. Simplified 49.6%

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

   if 1.54999999999999991e-205 < d < 5.4999999999999999e138

   1. Initial program 74.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 75.8%

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

      1. associate-*r* N/A

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

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

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

      3. sqrt-div N/A

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

      4. sqrt-div N/A

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

      5. frac-times N/A

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

      6. rem-square-sqrt N/A

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

      7. sqrt-prod N/A

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

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

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

      13. associate-/l* N/A

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

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

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

   10. Applied egg-rr 69.1%

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

   if 5.4999999999999999e138 < d

   1. Initial program 66.9%

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

   4. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-43}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-205}:\\ \;\;\;\;h \cdot \left(\sqrt{\frac{1}{\ell \cdot \left(h \cdot \left(\ell \cdot \ell\right)\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}}}{\frac{d \cdot 4}{D \cdot M}} \cdot \frac{h \cdot -0.5}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-201}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 + \frac{\frac{D\_m}{\frac{d}{M\_m}}}{\frac{d \cdot 4}{D\_m \cdot M\_m}} \cdot \frac{h \cdot -0.5}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \end{array} \]

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 -5.8e-201)
   (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -2e-310)
     (/ d (pow (* (* l h) (* l h)) 0.25))
     (if (<= l 4e+172)
       (*
        (/ d (pow (* l h) 0.5))
        (+
         1.0
         (* (/ (/ D_m (/ d M_m)) (/ (* d 4.0) (* D_m M_m))) (/ (* h -0.5) l))))
       (/ (* d (sqrt (/ h 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 <= -5.8e-201) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (l <= -2e-310) {
		tmp = d / pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 4e+172) {
		tmp = (d / pow((l * h), 0.5)) * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	} else {
		tmp = (d * sqrt((h / 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 <= (-5.8d-201)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (l <= (-2d-310)) then
        tmp = d / (((l * h) * (l * h)) ** 0.25d0)
    else if (l <= 4d+172) then
        tmp = (d / ((l * h) ** 0.5d0)) * (1.0d0 + (((d_m / (d / m_m)) / ((d * 4.0d0) / (d_m * m_m))) * ((h * (-0.5d0)) / l)))
    else
        tmp = (d * sqrt((h / 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 <= -5.8e-201) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -2e-310) {
		tmp = d / Math.pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 4e+172) {
		tmp = (d / Math.pow((l * h), 0.5)) * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	} else {
		tmp = (d * Math.sqrt((h / 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 <= -5.8e-201:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif l <= -2e-310:
		tmp = d / math.pow(((l * h) * (l * h)), 0.25)
	elif l <= 4e+172:
		tmp = (d / math.pow((l * h), 0.5)) * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)))
	else:
		tmp = (d * math.sqrt((h / 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 <= -5.8e-201)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -2e-310)
		tmp = Float64(d / (Float64(Float64(l * h) * Float64(l * h)) ^ 0.25));
	elseif (l <= 4e+172)
		tmp = Float64(Float64(d / (Float64(l * h) ^ 0.5)) * Float64(1.0 + Float64(Float64(Float64(D_m / Float64(d / M_m)) / Float64(Float64(d * 4.0) / Float64(D_m * M_m))) * Float64(Float64(h * -0.5) / l))));
	else
		tmp = Float64(Float64(d * sqrt(Float64(h / 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 <= -5.8e-201)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (l <= -2e-310)
		tmp = d / (((l * h) * (l * h)) ^ 0.25);
	elseif (l <= 4e+172)
		tmp = (d / ((l * h) ^ 0.5)) * (1.0 + (((D_m / (d / M_m)) / ((d * 4.0) / (D_m * M_m))) * ((h * -0.5) / l)));
	else
		tmp = (d * sqrt((h / 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, -5.8e-201], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+172], N[(N[(d / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.8000000000000003e-201

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. 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 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 43.1%

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

   if -5.8000000000000003e-201 < l < -1.999999999999994e-310

   1. Initial program 85.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 39.9%

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

   7. Simplified 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 39.9%

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

   9. Applied egg-rr 39.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\ell \cdot h\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{4}}}\right)\right) \]

       4. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right), \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \frac{1}{4}\right)\right) \]

   11. Applied egg-rr 58.1%

       \[\leadsto \frac{d}{\color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}} \]

   if -1.999999999999994e-310 < l < 4.0000000000000003e172

   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 58.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. 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 72.2%

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

   6. Applied egg-rr 72.2%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 76.5%

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

      1. associate-*r* N/A

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

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

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

      3. sqrt-div N/A

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

      4. sqrt-div N/A

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

      5. frac-times N/A

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

      6. rem-square-sqrt N/A

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

      7. sqrt-prod N/A

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

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

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

      13. associate-/l* N/A

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

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

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

   10. Applied egg-rr 79.0%

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

   if 4.0000000000000003e172 < l

   1. Initial program 56.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.0%

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

   5. Taylor expanded in h around 0

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

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

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

   7. Simplified 21.5%

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

   8. Taylor expanded in h around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. /-lowering-/.f64 52.8%

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

   10. Simplified 52.8%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-201}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\frac{d}{{\left(\ell \cdot h\right)}^{0.5}} \cdot \left(1 + \frac{\frac{D}{\frac{d}{M}}}{\frac{d \cdot 4}{D \cdot M}} \cdot \frac{h \cdot -0.5}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-206}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d}\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \end{array} \]

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 -1.25e-206)
   (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -2e-310)
     (/ d (pow (* (* l h) (* l h)) 0.25))
     (if (<= l 3.1e-32)
       (*
        (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 d))
        (sqrt (/ h (* l (* l l)))))
       (* d (sqrt (* (/ 1.0 h) (/ 1.0 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 <= -1.25e-206) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (l <= -2e-310) {
		tmp = d / pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 3.1e-32) {
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)) * sqrt((h / (l * (l * l))));
	} else {
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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 <= (-1.25d-206)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (l <= (-2d-310)) then
        tmp = d / (((l * h) * (l * h)) ** 0.25d0)
    else if (l <= 3.1d-32) then
        tmp = ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / d)) * sqrt((h / (l * (l * l))))
    else
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / 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 <= -1.25e-206) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -2e-310) {
		tmp = d / Math.pow(((l * h) * (l * h)), 0.25);
	} else if (l <= 3.1e-32) {
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)) * Math.sqrt((h / (l * (l * l))));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / 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 <= -1.25e-206:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif l <= -2e-310:
		tmp = d / math.pow(((l * h) * (l * h)), 0.25)
	elif l <= 3.1e-32:
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)) * math.sqrt((h / (l * (l * l))))
	else:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / 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 <= -1.25e-206)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -2e-310)
		tmp = Float64(d / (Float64(Float64(l * h) * Float64(l * h)) ^ 0.25));
	elseif (l <= 3.1e-32)
		tmp = Float64(Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / d)) * sqrt(Float64(h / Float64(l * Float64(l * l)))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / 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 <= -1.25e-206)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (l <= -2e-310)
		tmp = d / (((l * h) * (l * h)) ^ 0.25);
	elseif (l <= 3.1e-32)
		tmp = ((D_m * (D_m * (M_m * M_m))) * (-0.125 / d)) * sqrt((h / (l * (l * l))));
	else
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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, -1.25e-206], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.1e-32], N[(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] * N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.25e-206

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. 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 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 43.1%

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

   if -1.25e-206 < l < -1.999999999999994e-310

   1. Initial program 85.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 39.9%

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

   7. Simplified 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 39.9%

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

   9. Applied egg-rr 39.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\ell \cdot h\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{4}}}\right)\right) \]

       4. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(d, \left({\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right), \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\ell \cdot h\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \mathsf{*.f64}\left(\ell, h\right)\right), \frac{1}{4}\right)\right) \]

   11. Applied egg-rr 58.1%

       \[\leadsto \frac{d}{\color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}} \]

   if -1.999999999999994e-310 < l < 3.10000000000000011e-32

   1. Initial program 76.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 82.5%

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

   9. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. 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 52.9%

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

   11. Simplified 52.9%

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

   if 3.10000000000000011e-32 < l

   1. Initial program 62.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 50.9%

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

   7. Simplified 50.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 52.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(1, h\right)\right)\right)\right) \]

   9. Applied egg-rr 52.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-206}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{0.25}}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.6 \cdot 10^{-118}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\sqrt{\frac{1}{\ell \cdot \left(h \cdot \left(\ell \cdot \ell\right)\right)}} \cdot \left(\left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)\\ \end{array} \end{array} \]

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 2.6e-118)
   (/ (* d (sqrt (/ h l))) h)
   (*
    h
    (*
     (sqrt (/ 1.0 (* l (* h (* l l)))))
     (* (* D_m (* D_m (* M_m M_m))) (/ -0.125 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 (M_m <= 2.6e-118) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = h * (sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / 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 (m_m <= 2.6d-118) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = h * (sqrt((1.0d0 / (l * (h * (l * l))))) * ((d_m * (d_m * (m_m * m_m))) * ((-0.125d0) / 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 (M_m <= 2.6e-118) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = h * (Math.sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / 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 M_m <= 2.6e-118:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = h * (math.sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / 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 (M_m <= 2.6e-118)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(h * Float64(sqrt(Float64(1.0 / Float64(l * Float64(h * Float64(l * l))))) * Float64(Float64(D_m * Float64(D_m * Float64(M_m * M_m))) * Float64(-0.125 / 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 (M_m <= 2.6e-118)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = h * (sqrt((1.0 / (l * (h * (l * l))))) * ((D_m * (D_m * (M_m * M_m))) * (-0.125 / 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[M$95$m, 2.6e-118], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(h * N[(N[Sqrt[N[(1.0 / N[(l * N[(h * N[(l * l), $MachinePrecision]), $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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.6e-118

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 52.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{h}{\ell}}\right)\right), h\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right)\right), h\right) \]

      3. /-lowering-/.f64 44.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right)\right), h\right) \]

   10. Simplified 44.5%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

   if 2.6e-118 < M

   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 55.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)} \]

   7. Simplified 35.1%

      \[\leadsto \color{blue}{h \cdot \left(d \cdot \sqrt{\frac{1}{\left(\ell \cdot \left(h \cdot h\right)\right) \cdot h}} + \sqrt{\frac{1}{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot h\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)} \]

   8. Taylor expanded in d around 0

      \[\leadsto \mathsf{*.f64}\left(h, \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(h, \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot {\ell}^{3}}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(h, \left(\sqrt{\frac{1}{h \cdot {\ell}^{3}}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot {\ell}^{3}}\right)\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot {\ell}^{3}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \left(\left(\ell \cdot \ell\right) \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \left({\ell}^{2} \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\left(h \cdot {\ell}^{2}\right) \cdot \ell\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(h \cdot {\ell}^{2}\right), \ell\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left({\ell}^{2}\right)\right), \ell\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(\ell \cdot \ell\right)\right), \ell\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\ell, \ell\right)\right), \ell\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\ell, \ell\right)\right), \ell\right)\right)\right), \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{d}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\ell, \ell\right)\right), \ell\right)\right)\right), \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{-1}{8}}{d}\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\ell, \ell\right)\right), \ell\right)\right)\right), \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{\frac{\frac{-1}{8}}{d}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\ell, \ell\right)\right), \ell\right)\right)\right), \mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \color{blue}{\left(\frac{\frac{-1}{8}}{d}\right)}\right)\right)\right) \]

   10. Simplified 43.4%

       \[\leadsto h \cdot \color{blue}{\left(\sqrt{\frac{1}{\left(h \cdot \left(\ell \cdot \ell\right)\right) \cdot \ell}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-118}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\sqrt{\frac{1}{\ell \cdot \left(h \cdot \left(\ell \cdot \ell\right)\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - d\right) \cdot {\left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)}^{0.25}}{h}\\ \end{array} \end{array} \]

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 (<= M_m 1.25e-51)
   (/ (* d (sqrt (/ h l))) h)
   (/ (* (- 0.0 d) (pow (* (/ h l) (/ h l)) 0.25)) 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 <= 1.25e-51) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = ((0.0 - d) * pow(((h / l) * (h / l)), 0.25)) / 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 <= 1.25d-51) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = ((0.0d0 - d) * (((h / l) * (h / l)) ** 0.25d0)) / 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 <= 1.25e-51) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = ((0.0 - d) * Math.pow(((h / l) * (h / l)), 0.25)) / 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 <= 1.25e-51:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = ((0.0 - d) * math.pow(((h / l) * (h / l)), 0.25)) / 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 (M_m <= 1.25e-51)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(Float64(0.0 - d) * (Float64(Float64(h / l) * Float64(h / l)) ^ 0.25)) / 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 <= 1.25e-51)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = ((0.0 - d) * (((h / l) * (h / l)) ^ 0.25)) / 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[M$95$m, 1.25e-51], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(0.0 - d), $MachinePrecision] * N[Power[N[(N[(h / l), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000001e-51

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 53.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{h}{\ell}}\right)\right), h\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right)\right), h\right) \]

      3. /-lowering-/.f64 44.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right)\right), h\right) \]

   10. Simplified 44.1%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

   if 1.25000000000000001e-51 < M

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 53.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right), h\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)\right), h\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right)\right), h\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)\right), h\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{\ell}}\right), \left(-1 \cdot d\right)\right), h\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right), \left(-1 \cdot d\right)\right), h\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(-1 \cdot d\right)\right), h\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right), h\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(0 - d\right)\right), h\right) \]

      10. --lowering--.f64 17.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

   10. Simplified 17.0%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(0 - d\right)}}{h} \]
   11. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{h}{\ell}\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{h}{\ell}\right), \left(\frac{h}{\ell}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \left(\frac{h}{\ell}\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{/.f64}\left(h, \ell\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

       8. metadata-eval 26.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{/.f64}\left(h, \ell\right)\right), \frac{1}{4}\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

   12. Applied egg-rr 26.3%

       \[\leadsto \frac{\color{blue}{{\left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)}^{0.25}} \cdot \left(0 - d\right)}{h} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - d\right) \cdot {\left(\frac{h}{\ell} \cdot \frac{h}{\ell}\right)}^{0.25}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;M\_m \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot \left(0 - d \cdot d\right)}{d}}{h}\\ \end{array} \end{array} \]

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 (/ h l))))
   (if (<= M_m 1.25e-51) (/ (* d t_0) h) (/ (/ (* t_0 (- 0.0 (* d d))) d) 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((h / l));
	double tmp;
	if (M_m <= 1.25e-51) {
		tmp = (d * t_0) / h;
	} else {
		tmp = ((t_0 * (0.0 - (d * d))) / d) / 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((h / l))
    if (m_m <= 1.25d-51) then
        tmp = (d * t_0) / h
    else
        tmp = ((t_0 * (0.0d0 - (d * d))) / d) / 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((h / l));
	double tmp;
	if (M_m <= 1.25e-51) {
		tmp = (d * t_0) / h;
	} else {
		tmp = ((t_0 * (0.0 - (d * d))) / d) / 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((h / l))
	tmp = 0
	if M_m <= 1.25e-51:
		tmp = (d * t_0) / h
	else:
		tmp = ((t_0 * (0.0 - (d * d))) / d) / 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(h / l))
	tmp = 0.0
	if (M_m <= 1.25e-51)
		tmp = Float64(Float64(d * t_0) / h);
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(0.0 - Float64(d * d))) / d) / 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((h / l));
	tmp = 0.0;
	if (M_m <= 1.25e-51)
		tmp = (d * t_0) / h;
	else
		tmp = ((t_0 * (0.0 - (d * d))) / d) / 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[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 1.25e-51], N[(N[(d * t$95$0), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(t$95$0 * N[(0.0 - N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000001e-51

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 53.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{h}{\ell}}\right)\right), h\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right)\right), h\right) \]

      3. /-lowering-/.f64 44.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right)\right), h\right) \]

   10. Simplified 44.1%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

   if 1.25000000000000001e-51 < M

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 53.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right), h\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)\right), h\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right)\right), h\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)\right), h\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{\ell}}\right), \left(-1 \cdot d\right)\right), h\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right), \left(-1 \cdot d\right)\right), h\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(-1 \cdot d\right)\right), h\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right), h\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \left(0 - d\right)\right), h\right) \]

      10. --lowering--.f64 17.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right), \mathsf{\_.f64}\left(0, d\right)\right), h\right) \]

   10. Simplified 17.0%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \left(0 - d\right)}}{h} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)\right), h\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right), h\right) \]

       3. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(0 - d\right) \cdot \sqrt{\frac{h}{\ell}}\right), h\right) \]

       4. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 \cdot 0 - d \cdot d}{0 + d} \cdot \sqrt{\frac{h}{\ell}}\right), h\right) \]

       5. +-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 \cdot 0 - d \cdot d}{d} \cdot \sqrt{\frac{h}{\ell}}\right), h\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(0 \cdot 0 - d \cdot d\right) \cdot \sqrt{\frac{h}{\ell}}}{d}\right), h\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(0 \cdot 0 - d \cdot d\right) \cdot \sqrt{\frac{h}{\ell}}\right), d\right), h\right) \]

   12. Applied egg-rr 25.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(0 - d \cdot d\right) \cdot \sqrt{\frac{h}{\ell}}}{d}}}{h} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{h}{\ell}} \cdot \left(0 - d \cdot d\right)}{d}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-210}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \end{array} \]

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 -4.8e-210)
   (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
   (* d (sqrt (* (/ 1.0 h) (/ 1.0 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 <= -4.8e-210) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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 <= (-4.8d-210)) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / 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 <= -4.8e-210) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / 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 <= -4.8e-210:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / 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 <= -4.8e-210)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / 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 <= -4.8e-210)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	else
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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, -4.8e-210], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.80000000000000008e-210

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. 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 63.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(d, 4\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \frac{-1}{2}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{M \cdot D}{d \cdot 4} \cdot \frac{M \cdot D}{d}\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right) \]

   7. 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

         \[\leadsto \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      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-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \left(0 - \color{blue}{d}\right)\right) \]

      12. --lowering--.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{d}\right)\right) \]

   9. Simplified 43.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(0 - d\right)} \]

   if -4.80000000000000008e-210 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\left(h \cdot \ell\right)}^{-1}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\left(\ell \cdot h\right)}^{-1}\right)\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\ell}^{-1} \cdot {h}^{-1}\right)\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell} \cdot {h}^{-1}\right)\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(1, h\right)\right)\right)\right) \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-210}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-200}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \end{array} \]

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 -1.2e-200)
   (* (- 0.0 d) (sqrt (/ 1.0 (* l h))))
   (* d (sqrt (* (/ 1.0 h) (/ 1.0 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 <= -1.2e-200) {
		tmp = (0.0 - d) * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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 <= (-1.2d-200)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / 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 <= -1.2e-200) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / 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 <= -1.2e-200:
		tmp = (0.0 - d) * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / 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 <= -1.2e-200)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / 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 <= -1.2e-200)
		tmp = (0.0 - d) * sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / h) * (1.0 / 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, -1.2e-200], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.20000000000000001e-200

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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 l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 42.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -1.20000000000000001e-200 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\left(h \cdot \ell\right)}^{-1}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\left(\ell \cdot h\right)}^{-1}\right)\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left({\ell}^{-1} \cdot {h}^{-1}\right)\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell} \cdot {h}^{-1}\right)\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\ell}\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \left(\frac{1}{h}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \ell\right), \mathsf{/.f64}\left(1, h\right)\right)\right)\right) \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-200}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-211}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 -6.2e-211)
   (* (- 0.0 d) (sqrt (/ 1.0 (* l h))))
   (* d (sqrt (/ (/ 1.0 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 <= -6.2e-211) {
		tmp = (0.0 - d) * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / 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 <= (-6.2d-211)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / 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 <= -6.2e-211) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / 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 <= -6.2e-211:
		tmp = (0.0 - d) * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / 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 <= -6.2e-211)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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 <= -6.2e-211)
		tmp = (0.0 - d) * sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / 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, -6.2e-211], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.1999999999999999e-211

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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 l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      4. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 42.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if -6.1999999999999999e-211 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), h\right)\right)\right) \]

      4. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-211}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 -5.3e-208) (/ (* d (sqrt (/ h l))) h) (* d (sqrt (/ (/ 1.0 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 <= -5.3e-208) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = d * sqrt(((1.0 / 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 <= (-5.3d-208)) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = d * sqrt(((1.0d0 / 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 <= -5.3e-208) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d * Math.sqrt(((1.0 / 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 <= -5.3e-208:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = d * math.sqrt(((1.0 / 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 <= -5.3e-208)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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 <= -5.3e-208)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = d * sqrt(((1.0 / 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, -5.3e-208], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.29999999999999983e-208

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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 h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}\right), \color{blue}{h}\right) \]

   7. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{h \cdot \left(h \cdot h\right)}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   8. Taylor expanded in h around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}, h\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \left(\sqrt{\frac{h}{\ell}}\right)\right), h\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right)\right), h\right) \]

      3. /-lowering-/.f64 37.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right)\right), h\right) \]

   10. Simplified 37.5%

       \[\leadsto \frac{\color{blue}{d \cdot \sqrt{\frac{h}{\ell}}}}{h} \]

   if -5.29999999999999983e-208 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), h\right)\right)\right) \]

      4. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 -1.48e-201)
   (sqrt (* (/ d h) (/ d l)))
   (* d (sqrt (/ (/ 1.0 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 <= -1.48e-201) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * sqrt(((1.0 / 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 <= (-1.48d-201)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * sqrt(((1.0d0 / 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 <= -1.48e-201) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / 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 <= -1.48e-201:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.sqrt(((1.0 / 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 <= -1.48e-201)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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 <= -1.48e-201)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * sqrt(((1.0 / 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, -1.48e-201], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.47999999999999996e-201

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 6.2%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell \cdot h}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

      7. frac-times N/A

         \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

      8. sqrt-div N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

      9. sqrt-div N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      10. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

      14. /-lowering-/.f64 31.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   9. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -1.47999999999999996e-201 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{1}{\ell \cdot h}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\ell}}{h}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\ell}\right), h\right)\right)\right) \]

      4. /-lowering-/.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 43.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.48 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (<= l -6.5e-205) (sqrt (* (/ d h) (/ d 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 (l <= -6.5e-205) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = 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 (l <= (-6.5d-205)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = 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 (l <= -6.5e-205) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = 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 l <= -6.5e-205:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = 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 (l <= -6.5e-205)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = 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 (l <= -6.5e-205)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = 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[l, -6.5e-205], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.49999999999999956e-205

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 6.2%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell \cdot h}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

      7. frac-times N/A

         \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

      8. sqrt-div N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

      9. sqrt-div N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      10. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

      14. /-lowering-/.f64 31.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   9. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -6.49999999999999956e-205 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{d}\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\left(h \cdot \ell\right)}^{-1}}\right), d\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right), d\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\frac{-1}{2}}\right), d\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \frac{-1}{2}\right), d\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \frac{-1}{2}\right), d\right) \]

      8. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 42.9%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

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 (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) {
	return d * pow((l * h), -0.5);
}

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 * ((l * h) ** (-0.5d0))
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.pow((l * h), -0.5);
}

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.pow((l * h), -0.5)

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 * (Float64(l * h) ^ -0.5))
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 * ((l * h) ^ -0.5);
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[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 53.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

   4. *-lowering-*.f64 26.5%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

7. Simplified 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{h \cdot \ell}}\right), \color{blue}{d}\right) \]

   3. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{\left(h \cdot \ell\right)}^{-1}}\right), d\right) \]

   4. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right), d\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\frac{-1}{2}}\right), d\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \frac{-1}{2}\right), d\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \frac{-1}{2}\right), d\right) \]

   8. *-lowering-*.f64 26.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

9. Applied egg-rr 26.6%

   \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

10. Final simplification 26.6%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 23: 26.5% 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 67.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 53.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{4 \cdot \left(d \cdot d\right)} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right)\right) \]

   4. *-lowering-*.f64 26.5%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

7. Simplified 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   2. metadata-eval N/A

      \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

   7. *-lowering-*.f64 26.6%

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

9. Applied egg-rr 26.6%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))