Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 88.0%

Time: 29.9s

Alternatives: 28

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% 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: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{t\_0}} \cdot -0.125 + 1\\ \mathbf{if}\;h \leq -5.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot t\_1}{\frac{{\left(0 - h\right)}^{0.5}}{{\left(0 - d\right)}^{0.5}}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(h \cdot \left(t\_0 \cdot 0.25\right)\right) \cdot \frac{\frac{t\_0}{-2}}{\ell} + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l t_0)) -0.125) 1.0)))
   (if (<= h -5.2e+125)
     (/ (* (pow (/ d l) 0.5) t_1) (/ (pow (- 0.0 h) 0.5) (pow (- 0.0 d) 0.5)))
     (if (<= h -4e-310)
       (/ (* t_1 (/ (sqrt (- 0.0 d)) (pow (- 0.0 l) 0.5))) (sqrt (/ h d)))
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (/ (sqrt d) (sqrt h))
         (+ (* (* h (* t_0 0.25)) (/ (/ t_0 -2.0) l)) 1.0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (h <= -5.2e+125) {
		tmp = (pow((d / l), 0.5) * t_1) / (pow((0.0 - h), 0.5) / pow((0.0 - d), 0.5));
	} else if (h <= -4e-310) {
		tmp = (t_1 * (sqrt((0.0 - d)) / pow((0.0 - l), 0.5))) / sqrt((h / d));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * d_1) / d
    t_1 = (((h / (d / (m * d_1))) / (l / t_0)) * (-0.125d0)) + 1.0d0
    if (h <= (-5.2d+125)) then
        tmp = (((d / l) ** 0.5d0) * t_1) / (((0.0d0 - h) ** 0.5d0) / ((0.0d0 - d) ** 0.5d0))
    else if (h <= (-4d-310)) then
        tmp = (t_1 * (sqrt((0.0d0 - d)) / ((0.0d0 - l) ** 0.5d0))) / sqrt((h / d))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (((h * (t_0 * 0.25d0)) * ((t_0 / (-2.0d0)) / l)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (h <= -5.2e+125) {
		tmp = (Math.pow((d / l), 0.5) * t_1) / (Math.pow((0.0 - h), 0.5) / Math.pow((0.0 - d), 0.5));
	} else if (h <= -4e-310) {
		tmp = (t_1 * (Math.sqrt((0.0 - d)) / Math.pow((0.0 - l), 0.5))) / Math.sqrt((h / d));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * (((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) / d
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0
	tmp = 0
	if h <= -5.2e+125:
		tmp = (math.pow((d / l), 0.5) * t_1) / (math.pow((0.0 - h), 0.5) / math.pow((0.0 - d), 0.5))
	elif h <= -4e-310:
		tmp = (t_1 * (math.sqrt((0.0 - d)) / math.pow((0.0 - l), 0.5))) / math.sqrt((h / d))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * (((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / t_0)) * -0.125) + 1.0)
	tmp = 0.0
	if (h <= -5.2e+125)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * t_1) / Float64((Float64(0.0 - h) ^ 0.5) / (Float64(0.0 - d) ^ 0.5)));
	elseif (h <= -4e-310)
		tmp = Float64(Float64(t_1 * Float64(sqrt(Float64(0.0 - d)) / (Float64(0.0 - l) ^ 0.5))) / sqrt(Float64(h / d)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(Float64(h * Float64(t_0 * 0.25)) * Float64(Float64(t_0 / -2.0) / l)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) / d;
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	tmp = 0.0;
	if (h <= -5.2e+125)
		tmp = (((d / l) ^ 0.5) * t_1) / (((0.0 - h) ^ 0.5) / ((0.0 - d) ^ 0.5));
	elseif (h <= -4e-310)
		tmp = (t_1 * (sqrt((0.0 - d)) / ((0.0 - l) ^ 0.5))) / sqrt((h / d));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -5.2e+125], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Power[N[(0.0 - h), $MachinePrecision], 0.5], $MachinePrecision] / N[Power[N[(0.0 - d), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], N[(N[(t$95$1 * N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Power[N[(0.0 - l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / -2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.20000000000000006e125

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

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

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

      4. pow1/2 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(0 - h\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{d}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{d}\right)}\right)\right)\right) \]

      8. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \left({\left(\mathsf{neg}\left(d\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(d\right)\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(0 - d\right), \frac{1}{2}\right)\right)\right) \]

      11. --lowering--.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, d\right), \frac{1}{2}\right)\right)\right) \]

   8. Applied egg-rr 86.0%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{{\left(0 - h\right)}^{0.5}}{{\left(0 - d\right)}^{0.5}}}} \]

   if -5.20000000000000006e125 < h < -3.999999999999988e-310

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. metadata-eval 74.6%

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

   8. Applied egg-rr 74.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\ell}{d}\right)}^{-0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. sqrt-pow1 N/A

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

      3. inv-pow N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

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

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

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

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - d\right)\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \left({\left(\mathsf{neg}\left(\ell\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\left(0 - \ell\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      14. --lowering--.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

   10. Applied egg-rr 87.1%

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}} \]

   if -3.999999999999988e-310 < h

   1. Initial program 63.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-div N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 77.1%

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

   6. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. sqrt-div N/A

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

      2. pow1/2 N/A

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

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

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

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

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

      5. pow1/2 N/A

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

      6. sqrt-lowering-sqrt.f64 87.1%

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

   8. Applied egg-rr 87.1%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\frac{{\left(0 - h\right)}^{0.5}}{{\left(0 - d\right)}^{0.5}}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell} + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{t\_0}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -3.15 \cdot 10^{-229}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot t\_1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;t\_1 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(t\_0 \cdot 0.25\right)\right) \cdot \frac{\frac{t\_0}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l t_0)) -0.125) 1.0)))
   (if (<= d -3.15e-229)
     (/ (* (pow (/ d l) 0.5) t_1) (sqrt (/ h d)))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 1.35e-66)
         (* t_1 (/ d (pow (* h l) 0.5)))
         (*
          (/ (sqrt d) (sqrt l))
          (*
           (+ (* (* h (* t_0 0.25)) (/ (/ t_0 -2.0) l)) 1.0)
           (sqrt (/ d h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (d <= -3.15e-229) {
		tmp = (pow((d / l), 0.5) * t_1) / sqrt((h / d));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 1.35e-66) {
		tmp = t_1 * (d / pow((h * l), 0.5));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * d_1) / d
    t_1 = (((h / (d / (m * d_1))) / (l / t_0)) * (-0.125d0)) + 1.0d0
    if (d <= (-3.15d-229)) then
        tmp = (((d / l) ** 0.5d0) * t_1) / sqrt((h / d))
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 1.35d-66) then
        tmp = t_1 * (d / ((h * l) ** 0.5d0))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25d0)) * ((t_0 / (-2.0d0)) / l)) + 1.0d0) * sqrt((d / h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (d <= -3.15e-229) {
		tmp = (Math.pow((d / l), 0.5) * t_1) / Math.sqrt((h / d));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 1.35e-66) {
		tmp = t_1 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * Math.sqrt((d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) / d
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0
	tmp = 0
	if d <= -3.15e-229:
		tmp = (math.pow((d / l), 0.5) * t_1) / math.sqrt((h / d))
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 1.35e-66:
		tmp = t_1 * (d / math.pow((h * l), 0.5))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * math.sqrt((d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / t_0)) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -3.15e-229)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * t_1) / sqrt(Float64(h / d)));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 1.35e-66)
		tmp = Float64(t_1 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(Float64(Float64(h * Float64(t_0 * 0.25)) * Float64(Float64(t_0 / -2.0) / l)) + 1.0) * sqrt(Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) / d;
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -3.15e-229)
		tmp = (((d / l) ^ 0.5) * t_1) / sqrt((h / d));
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 1.35e-66)
		tmp = t_1 * (d / ((h * l) ^ 0.5));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -3.15e-229], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-66], N[(t$95$1 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / -2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.14999999999999993e-229

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 80.4%

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

   if -3.14999999999999993e-229 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 1.34999999999999998e-66

   1. Initial program 42.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.9%

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

   if 1.34999999999999998e-66 < d

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 81.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-div N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 88.0%

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

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.15 \cdot 10^{-229}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ t_1 := \sqrt{\frac{h}{d}}\\ t_2 := \frac{\frac{h}{t\_0}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot t\_2}{t\_1}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;t\_2 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \left(\frac{\frac{h \cdot -0.125}{t\_0}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right)}{\sqrt{\ell}}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D)))
        (t_1 (sqrt (/ h d)))
        (t_2 (+ (* (/ (/ h t_0) (/ l (/ (* M D) d))) -0.125) 1.0)))
   (if (<= d -1.3e-230)
     (/ (* (pow (/ d l) 0.5) t_2) t_1)
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 6.2e-57)
         (* t_2 (/ d (pow (* h l) 0.5)))
         (/
          (/
           (* (sqrt d) (+ (/ (/ (* h -0.125) t_0) (/ (/ l M) (/ D d))) 1.0))
           (sqrt l))
          t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = sqrt((h / d));
	double t_2 = (((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.3e-230) {
		tmp = (pow((d / l), 0.5) * t_2) / t_1;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 6.2e-57) {
		tmp = t_2 * (d / pow((h * l), 0.5));
	} else {
		tmp = ((sqrt(d) * ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0)) / sqrt(l)) / t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d / (m * d_1)
    t_1 = sqrt((h / d))
    t_2 = (((h / t_0) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0
    if (d <= (-1.3d-230)) then
        tmp = (((d / l) ** 0.5d0) * t_2) / t_1
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 6.2d-57) then
        tmp = t_2 * (d / ((h * l) ** 0.5d0))
    else
        tmp = ((sqrt(d) * ((((h * (-0.125d0)) / t_0) / ((l / m) / (d_1 / d))) + 1.0d0)) / sqrt(l)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = Math.sqrt((h / d));
	double t_2 = (((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.3e-230) {
		tmp = (Math.pow((d / l), 0.5) * t_2) / t_1;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 6.2e-57) {
		tmp = t_2 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = ((Math.sqrt(d) * ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0)) / Math.sqrt(l)) / t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	t_1 = math.sqrt((h / d))
	t_2 = (((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0
	tmp = 0
	if d <= -1.3e-230:
		tmp = (math.pow((d / l), 0.5) * t_2) / t_1
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 6.2e-57:
		tmp = t_2 * (d / math.pow((h * l), 0.5))
	else:
		tmp = ((math.sqrt(d) * ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0)) / math.sqrt(l)) / t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	t_1 = sqrt(Float64(h / d))
	t_2 = Float64(Float64(Float64(Float64(h / t_0) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -1.3e-230)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * t_2) / t_1);
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 6.2e-57)
		tmp = Float64(t_2 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) * Float64(Float64(Float64(Float64(h * -0.125) / t_0) / Float64(Float64(l / M) / Float64(D / d))) + 1.0)) / sqrt(l)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	t_1 = sqrt((h / d));
	t_2 = (((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -1.3e-230)
		tmp = (((d / l) ^ 0.5) * t_2) / t_1;
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 6.2e-57)
		tmp = t_2 * (d / ((h * l) ^ 0.5));
	else
		tmp = ((sqrt(d) * ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0)) / sqrt(l)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.3e-230], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.2e-57], N[(t$95$2 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[(h * -0.125), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(l / M), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3000000000000001e-230

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 80.4%

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

   if -1.3000000000000001e-230 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 6.19999999999999952e-57

   1. Initial program 42.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.0%

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

   if 6.19999999999999952e-57 < d

   1. Initial program 78.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 82.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. sqrt-div N/A

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

      4. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 86.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\frac{-0.125 \cdot h}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{d}}{\sqrt{\ell}}}}{\sqrt{\frac{h}{d}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \left(\frac{\frac{h \cdot -0.125}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right)}{\sqrt{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{d}}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -2.85 \cdot 10^{-229}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot t\_1}{t\_0}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-57}:\\ \;\;\;\;t\_1 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \frac{t\_1}{t\_0}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h d)))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l (/ (* M D) d))) -0.125) 1.0)))
   (if (<= d -2.85e-229)
     (/ (* (pow (/ d l) 0.5) t_1) t_0)
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 6.5e-57)
         (* t_1 (/ d (pow (* h l) 0.5)))
         (/ (* (sqrt d) (/ t_1 t_0)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / d));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -2.85e-229) {
		tmp = (pow((d / l), 0.5) * t_1) / t_0;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 6.5e-57) {
		tmp = t_1 * (d / pow((h * l), 0.5));
	} else {
		tmp = (sqrt(d) * (t_1 / t_0)) / sqrt(l);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h / d))
    t_1 = (((h / (d / (m * d_1))) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0
    if (d <= (-2.85d-229)) then
        tmp = (((d / l) ** 0.5d0) * t_1) / t_0
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 6.5d-57) then
        tmp = t_1 * (d / ((h * l) ** 0.5d0))
    else
        tmp = (sqrt(d) * (t_1 / t_0)) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / d));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -2.85e-229) {
		tmp = (Math.pow((d / l), 0.5) * t_1) / t_0;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 6.5e-57) {
		tmp = t_1 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (Math.sqrt(d) * (t_1 / t_0)) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / d))
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0
	tmp = 0
	if d <= -2.85e-229:
		tmp = (math.pow((d / l), 0.5) * t_1) / t_0
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 6.5e-57:
		tmp = t_1 * (d / math.pow((h * l), 0.5))
	else:
		tmp = (math.sqrt(d) * (t_1 / t_0)) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / d))
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -2.85e-229)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * t_1) / t_0);
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 6.5e-57)
		tmp = Float64(t_1 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(t_1 / t_0)) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / d));
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -2.85e-229)
		tmp = (((d / l) ^ 0.5) * t_1) / t_0;
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 6.5e-57)
		tmp = t_1 * (d / ((h * l) ^ 0.5));
	else
		tmp = (sqrt(d) * (t_1 / t_0)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -2.85e-229], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e-57], N[(t$95$1 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.85000000000000012e-229

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 80.4%

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

   if -2.85000000000000012e-229 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 6.49999999999999992e-57

   1. Initial program 42.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.0%

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

   if 6.49999999999999992e-57 < d

   1. Initial program 78.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{d}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{-229}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-57}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \frac{\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1}{\sqrt{\frac{h}{d}}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{t\_0}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-62}:\\ \;\;\;\;t\_1 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(t\_0 \cdot 0.25\right)\right) \cdot \frac{\frac{t\_0}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l t_0)) -0.125) 1.0)))
   (if (<= d -3.8e-299)
     (/ (* t_1 (/ (sqrt (- 0.0 d)) (pow (- 0.0 l) 0.5))) (sqrt (/ h d)))
     (if (<= d 2.1e-62)
       (* t_1 (/ d (pow (* h l) 0.5)))
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (+ (* (* h (* t_0 0.25)) (/ (/ t_0 -2.0) l)) 1.0)
         (sqrt (/ d h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (d <= -3.8e-299) {
		tmp = (t_1 * (sqrt((0.0 - d)) / pow((0.0 - l), 0.5))) / sqrt((h / d));
	} else if (d <= 2.1e-62) {
		tmp = t_1 * (d / pow((h * l), 0.5));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * d_1) / d
    t_1 = (((h / (d / (m * d_1))) / (l / t_0)) * (-0.125d0)) + 1.0d0
    if (d <= (-3.8d-299)) then
        tmp = (t_1 * (sqrt((0.0d0 - d)) / ((0.0d0 - l) ** 0.5d0))) / sqrt((h / d))
    else if (d <= 2.1d-62) then
        tmp = t_1 * (d / ((h * l) ** 0.5d0))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25d0)) * ((t_0 / (-2.0d0)) / l)) + 1.0d0) * sqrt((d / h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	double tmp;
	if (d <= -3.8e-299) {
		tmp = (t_1 * (Math.sqrt((0.0 - d)) / Math.pow((0.0 - l), 0.5))) / Math.sqrt((h / d));
	} else if (d <= 2.1e-62) {
		tmp = t_1 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * Math.sqrt((d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) / d
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0
	tmp = 0
	if d <= -3.8e-299:
		tmp = (t_1 * (math.sqrt((0.0 - d)) / math.pow((0.0 - l), 0.5))) / math.sqrt((h / d))
	elif d <= 2.1e-62:
		tmp = t_1 * (d / math.pow((h * l), 0.5))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * math.sqrt((d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / t_0)) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -3.8e-299)
		tmp = Float64(Float64(t_1 * Float64(sqrt(Float64(0.0 - d)) / (Float64(0.0 - l) ^ 0.5))) / sqrt(Float64(h / d)));
	elseif (d <= 2.1e-62)
		tmp = Float64(t_1 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(Float64(Float64(h * Float64(t_0 * 0.25)) * Float64(Float64(t_0 / -2.0) / l)) + 1.0) * sqrt(Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) / d;
	t_1 = (((h / (d / (M * D))) / (l / t_0)) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -3.8e-299)
		tmp = (t_1 * (sqrt((0.0 - d)) / ((0.0 - l) ^ 0.5))) / sqrt((h / d));
	elseif (d <= 2.1e-62)
		tmp = t_1 * (d / ((h * l) ^ 0.5));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_0 * 0.25)) * ((t_0 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -3.8e-299], N[(N[(t$95$1 * N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Power[N[(0.0 - l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.1e-62], N[(t$95$1 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / -2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.8000000000000003e-299

   1. Initial program 69.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 74.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\ell}{d}\right)}^{-0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. sqrt-pow1 N/A

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

      3. inv-pow N/A

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

      4. clear-num N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

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

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

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

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - d\right)\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \left(\sqrt{\mathsf{neg}\left(\ell\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \left({\left(\mathsf{neg}\left(\ell\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\left(0 - \ell\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      14. --lowering--.f64 84.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \frac{1}{2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

   10. Applied egg-rr 84.2%

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}} \]

   if -3.8000000000000003e-299 < d < 2.0999999999999999e-62

   1. Initial program 40.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 70.9%

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

   if 2.0999999999999999e-62 < d

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 81.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-div N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 88.0%

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

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{\sqrt{0 - d}}{{\left(0 - \ell\right)}^{0.5}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-62}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ t_1 := \frac{M \cdot D}{d}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(\frac{\frac{h \cdot -0.125}{t\_0}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\frac{{\left(0 - h\right)}^{0.5}}{\sqrt{0 - d}}}\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{-68}:\\ \;\;\;\;\left(\frac{\frac{h}{t\_0}}{\frac{\ell}{t\_1}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(t\_1 \cdot 0.25\right)\right) \cdot \frac{\frac{t\_1}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D))) (t_1 (/ (* M D) d)))
   (if (<= d -3.2e-302)
     (/
      (* (+ (/ (/ (* h -0.125) t_0) (/ (/ l M) (/ D d))) 1.0) (sqrt (/ d l)))
      (/ (pow (- 0.0 h) 0.5) (sqrt (- 0.0 d))))
     (if (<= d 5.7e-68)
       (* (+ (* (/ (/ h t_0) (/ l t_1)) -0.125) 1.0) (/ d (pow (* h l) 0.5)))
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (+ (* (* h (* t_1 0.25)) (/ (/ t_1 -2.0) l)) 1.0)
         (sqrt (/ d h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = (M * D) / d;
	double tmp;
	if (d <= -3.2e-302) {
		tmp = (((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * sqrt((d / l))) / (pow((0.0 - h), 0.5) / sqrt((0.0 - d)));
	} else if (d <= 5.7e-68) {
		tmp = ((((h / t_0) / (l / t_1)) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_1 * 0.25)) * ((t_1 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / (m * d_1)
    t_1 = (m * d_1) / d
    if (d <= (-3.2d-302)) then
        tmp = (((((h * (-0.125d0)) / t_0) / ((l / m) / (d_1 / d))) + 1.0d0) * sqrt((d / l))) / (((0.0d0 - h) ** 0.5d0) / sqrt((0.0d0 - d)))
    else if (d <= 5.7d-68) then
        tmp = ((((h / t_0) / (l / t_1)) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_1 * 0.25d0)) * ((t_1 / (-2.0d0)) / l)) + 1.0d0) * sqrt((d / h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = (M * D) / d;
	double tmp;
	if (d <= -3.2e-302) {
		tmp = (((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * Math.sqrt((d / l))) / (Math.pow((0.0 - h), 0.5) / Math.sqrt((0.0 - d)));
	} else if (d <= 5.7e-68) {
		tmp = ((((h / t_0) / (l / t_1)) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((((h * (t_1 * 0.25)) * ((t_1 / -2.0) / l)) + 1.0) * Math.sqrt((d / h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	t_1 = (M * D) / d
	tmp = 0
	if d <= -3.2e-302:
		tmp = (((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * math.sqrt((d / l))) / (math.pow((0.0 - h), 0.5) / math.sqrt((0.0 - d)))
	elif d <= 5.7e-68:
		tmp = ((((h / t_0) / (l / t_1)) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((((h * (t_1 * 0.25)) * ((t_1 / -2.0) / l)) + 1.0) * math.sqrt((d / h)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	t_1 = Float64(Float64(M * D) / d)
	tmp = 0.0
	if (d <= -3.2e-302)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(h * -0.125) / t_0) / Float64(Float64(l / M) / Float64(D / d))) + 1.0) * sqrt(Float64(d / l))) / Float64((Float64(0.0 - h) ^ 0.5) / sqrt(Float64(0.0 - d))));
	elseif (d <= 5.7e-68)
		tmp = Float64(Float64(Float64(Float64(Float64(h / t_0) / Float64(l / t_1)) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(Float64(Float64(h * Float64(t_1 * 0.25)) * Float64(Float64(t_1 / -2.0) / l)) + 1.0) * sqrt(Float64(d / h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	t_1 = (M * D) / d;
	tmp = 0.0;
	if (d <= -3.2e-302)
		tmp = (((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * sqrt((d / l))) / (((0.0 - h) ^ 0.5) / sqrt((0.0 - d)));
	elseif (d <= 5.7e-68)
		tmp = ((((h / t_0) / (l / t_1)) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((((h * (t_1 * 0.25)) * ((t_1 / -2.0) / l)) + 1.0) * sqrt((d / h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.2e-302], N[(N[(N[(N[(N[(N[(h * -0.125), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(l / M), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(0.0 - h), $MachinePrecision], 0.5], $MachinePrecision] / N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.7e-68], N[(N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * N[(t$95$1 * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / -2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.19999999999999978e-302

   1. Initial program 68.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 73.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

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

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

   8. Applied egg-rr 71.4%

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

      1. frac-2neg N/A

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

      2. sqrt-div N/A

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

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

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

      4. pow1/2 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, h\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, M\right), \mathsf{/.f64}\left(D, d\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(0 - h\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{d}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, h\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, M\right), \mathsf{/.f64}\left(D, d\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \left(\sqrt{\mathsf{neg}\left(\color{blue}{d}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, h\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, M\right), \mathsf{/.f64}\left(D, d\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, h\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, M\right), \mathsf{/.f64}\left(D, d\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(0 - d\right)\right)\right)\right) \]

      10. --lowering--.f64 80.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, h\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, M\right), \mathsf{/.f64}\left(D, d\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, h\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, d\right)\right)\right)\right) \]

   10. Applied egg-rr 80.2%

       \[\leadsto \frac{\left(\frac{\frac{-0.125 \cdot h}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{\frac{{\left(0 - h\right)}^{0.5}}{\sqrt{0 - d}}}} \]

   if -3.19999999999999978e-302 < d < 5.7000000000000002e-68

   1. Initial program 42.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.9%

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

   if 5.7000000000000002e-68 < d

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 81.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-div N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 88.0%

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

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(\frac{\frac{h \cdot -0.125}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\frac{{\left(0 - h\right)}^{0.5}}{\sqrt{0 - d}}}\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{-68}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell} + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{d}}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -1.08 \cdot 10^{-228}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot t\_1}{t\_0}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;t\_1 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h d)))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l (/ (* M D) d))) -0.125) 1.0)))
   (if (<= d -1.08e-228)
     (/ (* (pow (/ d l) 0.5) t_1) t_0)
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 7.5e-89)
         (* t_1 (/ d (pow (* h l) 0.5)))
         (/
          (*
           (sqrt (/ d l))
           (+ (* (* h -0.125) (* (/ M (/ d D)) (/ D (* d (/ l M))))) 1.0))
          t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / d));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.08e-228) {
		tmp = (pow((d / l), 0.5) * t_1) / t_0;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 7.5e-89) {
		tmp = t_1 * (d / pow((h * l), 0.5));
	} else {
		tmp = (sqrt((d / l)) * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h / d))
    t_1 = (((h / (d / (m * d_1))) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0
    if (d <= (-1.08d-228)) then
        tmp = (((d / l) ** 0.5d0) * t_1) / t_0
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 7.5d-89) then
        tmp = t_1 * (d / ((h * l) ** 0.5d0))
    else
        tmp = (sqrt((d / l)) * (((h * (-0.125d0)) * ((m / (d / d_1)) * (d_1 / (d * (l / m))))) + 1.0d0)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / d));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.08e-228) {
		tmp = (Math.pow((d / l), 0.5) * t_1) / t_0;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 7.5e-89) {
		tmp = t_1 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (Math.sqrt((d / l)) * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / d))
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0
	tmp = 0
	if d <= -1.08e-228:
		tmp = (math.pow((d / l), 0.5) * t_1) / t_0
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 7.5e-89:
		tmp = t_1 * (d / math.pow((h * l), 0.5))
	else:
		tmp = (math.sqrt((d / l)) * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / d))
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -1.08e-228)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * t_1) / t_0);
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 7.5e-89)
		tmp = Float64(t_1 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(Float64(Float64(h * -0.125) * Float64(Float64(M / Float64(d / D)) * Float64(D / Float64(d * Float64(l / M))))) + 1.0)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / d));
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -1.08e-228)
		tmp = (((d / l) ^ 0.5) * t_1) / t_0;
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 7.5e-89)
		tmp = t_1 * (d / ((h * l) ^ 0.5));
	else
		tmp = (sqrt((d / l)) * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.08e-228], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e-89], N[(t$95$1 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.0799999999999999e-228

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 80.4%

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

   if -1.0799999999999999e-228 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 7.4999999999999999e-89

   1. Initial program 35.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 70.9%

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

   if 7.4999999999999999e-89 < d

   1. Initial program 79.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

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

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

   8. Applied egg-rr 81.6%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

      15. clear-num N/A

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

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

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

      17. associate-/l/ N/A

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

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

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

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

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

      20. /-lowering-/.f64 82.7%

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

   10. Applied egg-rr 82.7%

       \[\leadsto \frac{\left(\color{blue}{\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{\ell}{M} \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.08 \cdot 10^{-228}:\\ \;\;\;\;\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-89}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{d}}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{-228}:\\ \;\;\;\;\frac{t\_1 \cdot \left(\frac{\frac{h \cdot -0.125}{t\_0}}{\frac{\ell \cdot \frac{d}{M}}{D}} + 1\right)}{t\_2}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{\frac{h}{t\_0}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D))) (t_1 (sqrt (/ d l))) (t_2 (sqrt (/ h d))))
   (if (<= d -1.75e-228)
     (/ (* t_1 (+ (/ (/ (* h -0.125) t_0) (/ (* l (/ d M)) D)) 1.0)) t_2)
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 4.2e-88)
         (*
          (+ (* (/ (/ h t_0) (/ l (/ (* M D) d))) -0.125) 1.0)
          (/ d (pow (* h l) 0.5)))
         (/
          (*
           t_1
           (+ (* (* h -0.125) (* (/ M (/ d D)) (/ D (* d (/ l M))))) 1.0))
          t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / d));
	double tmp;
	if (d <= -1.75e-228) {
		tmp = (t_1 * ((((h * -0.125) / t_0) / ((l * (d / M)) / D)) + 1.0)) / t_2;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.2e-88) {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	} else {
		tmp = (t_1 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d / (m * d_1)
    t_1 = sqrt((d / l))
    t_2 = sqrt((h / d))
    if (d <= (-1.75d-228)) then
        tmp = (t_1 * ((((h * (-0.125d0)) / t_0) / ((l * (d / m)) / d_1)) + 1.0d0)) / t_2
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 4.2d-88) then
        tmp = ((((h / t_0) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    else
        tmp = (t_1 * (((h * (-0.125d0)) * ((m / (d / d_1)) * (d_1 / (d * (l / m))))) + 1.0d0)) / t_2
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((h / d));
	double tmp;
	if (d <= -1.75e-228) {
		tmp = (t_1 * ((((h * -0.125) / t_0) / ((l * (d / M)) / D)) + 1.0)) / t_2;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.2e-88) {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (t_1 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((h / d))
	tmp = 0
	if d <= -1.75e-228:
		tmp = (t_1 * ((((h * -0.125) / t_0) / ((l * (d / M)) / D)) + 1.0)) / t_2
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 4.2e-88:
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	else:
		tmp = (t_1 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / d))
	tmp = 0.0
	if (d <= -1.75e-228)
		tmp = Float64(Float64(t_1 * Float64(Float64(Float64(Float64(h * -0.125) / t_0) / Float64(Float64(l * Float64(d / M)) / D)) + 1.0)) / t_2);
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 4.2e-88)
		tmp = Float64(Float64(Float64(Float64(Float64(h / t_0) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(t_1 * Float64(Float64(Float64(h * -0.125) * Float64(Float64(M / Float64(d / D)) * Float64(D / Float64(d * Float64(l / M))))) + 1.0)) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	t_1 = sqrt((d / l));
	t_2 = sqrt((h / d));
	tmp = 0.0;
	if (d <= -1.75e-228)
		tmp = (t_1 * ((((h * -0.125) / t_0) / ((l * (d / M)) / D)) + 1.0)) / t_2;
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 4.2e-88)
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	else
		tmp = (t_1 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.75e-228], N[(N[(t$95$1 * N[(N[(N[(N[(h * -0.125), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(l * N[(d / M), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-88], N[(N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.74999999999999987e-228

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

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

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

   8. Applied egg-rr 77.8%

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

      1. associate-/r/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. /-lowering-/.f64 77.9%

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

   10. Applied egg-rr 77.9%

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

   if -1.74999999999999987e-228 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 4.1999999999999999e-88

   1. Initial program 35.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 70.9%

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

   if 4.1999999999999999e-88 < d

   1. Initial program 79.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

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

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

   8. Applied egg-rr 81.6%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

      15. clear-num N/A

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

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

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

      17. associate-/l/ N/A

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

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

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

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

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

      20. /-lowering-/.f64 82.7%

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

   10. Applied egg-rr 82.7%

       \[\leadsto \frac{\left(\color{blue}{\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{\ell}{M} \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\frac{h \cdot -0.125}{\frac{d}{M \cdot D}}}{\frac{\ell \cdot \frac{d}{M}}{D}} + 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{-229}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{-53}:\\ \;\;\;\;t\_1 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (+ (* (/ (/ h (/ d (* M D))) (/ l (/ (* M D) d))) -0.125) 1.0)))
   (if (<= d -1.75e-229)
     (* t_0 (* t_1 (sqrt (/ d h))))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 1.04e-53)
         (* t_1 (/ d (pow (* h l) 0.5)))
         (/
          (*
           t_0
           (+ (* (* h -0.125) (* (/ M (/ d D)) (/ D (* d (/ l M))))) 1.0))
          (sqrt (/ h d))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.75e-229) {
		tmp = t_0 * (t_1 * sqrt((d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 1.04e-53) {
		tmp = t_1 * (d / pow((h * l), 0.5));
	} else {
		tmp = (t_0 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / sqrt((h / d));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (((h / (d / (m * d_1))) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0
    if (d <= (-1.75d-229)) then
        tmp = t_0 * (t_1 * sqrt((d / h)))
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 1.04d-53) then
        tmp = t_1 * (d / ((h * l) ** 0.5d0))
    else
        tmp = (t_0 * (((h * (-0.125d0)) * ((m / (d / d_1)) * (d_1 / (d * (l / m))))) + 1.0d0)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.75e-229) {
		tmp = t_0 * (t_1 * Math.sqrt((d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 1.04e-53) {
		tmp = t_1 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = (t_0 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0
	tmp = 0
	if d <= -1.75e-229:
		tmp = t_0 * (t_1 * math.sqrt((d / h)))
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 1.04e-53:
		tmp = t_1 * (d / math.pow((h * l), 0.5))
	else:
		tmp = (t_0 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / math.sqrt((h / d))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -1.75e-229)
		tmp = Float64(t_0 * Float64(t_1 * sqrt(Float64(d / h))));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 1.04e-53)
		tmp = Float64(t_1 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(Float64(h * -0.125) * Float64(Float64(M / Float64(d / D)) * Float64(D / Float64(d * Float64(l / M))))) + 1.0)) / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -1.75e-229)
		tmp = t_0 * (t_1 * sqrt((d / h)));
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 1.04e-53)
		tmp = t_1 * (d / ((h * l) ^ 0.5));
	else
		tmp = (t_0 * (((h * -0.125) * ((M / (d / D)) * (D / (d * (l / M))))) + 1.0)) / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.75e-229], N[(t$95$0 * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.04e-53], N[(t$95$1 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(N[(h * -0.125), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.7500000000000002e-229

   1. Initial program 74.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. unpow1/2 N/A

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

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

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

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

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

      13. sub-neg N/A

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

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

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-*r* N/A

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

      4. associate-/r/ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

   6. Applied egg-rr 80.4%

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

   if -1.7500000000000002e-229 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/r/ N/A

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

      9. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-commutative N/A

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

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 30.6%

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

       1. mul-1-neg N/A

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

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 1.04000000000000001e-53

   1. Initial program 41.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]

   if 1.04000000000000001e-53 < d

   1. Initial program 79.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 83.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, d\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, d\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right), \left(\sqrt{\frac{d}{\ell}}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(h, d\right)}\right)\right) \]

   8. Applied egg-rr 82.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{-0.125 \cdot h}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}}{\sqrt{\frac{h}{d}}} \]
   9. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot h}{\frac{d}{M \cdot D}} \cdot \frac{1}{\frac{\frac{\ell}{M}}{\frac{D}{d}}}\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{8} \cdot h\right) \cdot \frac{1}{\frac{d}{M \cdot D}}\right) \cdot \frac{1}{\frac{\frac{\ell}{M}}{\frac{D}{d}}}\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{8} \cdot h\right) \cdot \frac{M \cdot D}{d}\right) \cdot \frac{1}{\frac{\frac{\ell}{M}}{\frac{D}{d}}}\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{8} \cdot h\right) \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\frac{D}{d}}{\frac{\ell}{M}}\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{-1}{8} \cdot h\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot h\right), \left(\frac{M \cdot D}{d} \cdot \frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \frac{-1}{8}\right), \left(\frac{M \cdot D}{d} \cdot \frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \left(\frac{M \cdot D}{d} \cdot \frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(M \cdot \frac{D}{d}\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(M \cdot \frac{1}{\frac{1}{\frac{D}{d}}}\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(\frac{M}{\frac{1}{\frac{D}{d}}}\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{1}{\frac{D}{d}}\right)\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      15. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\frac{D}{d}}{\frac{\ell}{M}}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      17. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{D}{\frac{\ell}{M} \cdot d}\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(D, \left(\frac{\ell}{M} \cdot d\right)\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{\ell}{M}\right), d\right)\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      20. /-lowering-/.f64 83.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, M\right), d\right)\right)\right)\right), 1\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

   10. Applied egg-rr 83.5%

       \[\leadsto \frac{\left(\color{blue}{\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{\ell}{M} \cdot d}\right)} + 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{-53}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(h \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{d \cdot \frac{\ell}{M}}\right) + 1\right)}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \frac{\ell}{t\_0}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \frac{\frac{h}{\frac{d}{M \cdot D}}}{t\_1} \cdot -0.125 + 1\\ \mathbf{if}\;d \leq -1.62 \cdot 10^{-230}:\\ \;\;\;\;t\_2 \cdot \left(t\_4 \cdot t\_3\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-88}:\\ \;\;\;\;t\_4 \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \left(h \cdot \left(-0.125 \cdot \frac{t\_0}{t\_1}\right) + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (/ l t_0))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (/ d h)))
        (t_4 (+ (* (/ (/ h (/ d (* M D))) t_1) -0.125) 1.0)))
   (if (<= d -1.62e-230)
     (* t_2 (* t_4 t_3))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 4.1e-88)
         (* t_4 (/ d (pow (* h l) 0.5)))
         (* t_2 (* t_3 (+ (* h (* -0.125 (/ t_0 t_1))) 1.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = l / t_0;
	double t_2 = sqrt((d / l));
	double t_3 = sqrt((d / h));
	double t_4 = (((h / (d / (M * D))) / t_1) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.62e-230) {
		tmp = t_2 * (t_4 * t_3);
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.1e-88) {
		tmp = t_4 * (d / pow((h * l), 0.5));
	} else {
		tmp = t_2 * (t_3 * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (m * d_1) / d
    t_1 = l / t_0
    t_2 = sqrt((d / l))
    t_3 = sqrt((d / h))
    t_4 = (((h / (d / (m * d_1))) / t_1) * (-0.125d0)) + 1.0d0
    if (d <= (-1.62d-230)) then
        tmp = t_2 * (t_4 * t_3)
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 4.1d-88) then
        tmp = t_4 * (d / ((h * l) ** 0.5d0))
    else
        tmp = t_2 * (t_3 * ((h * ((-0.125d0) * (t_0 / t_1))) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = l / t_0;
	double t_2 = Math.sqrt((d / l));
	double t_3 = Math.sqrt((d / h));
	double t_4 = (((h / (d / (M * D))) / t_1) * -0.125) + 1.0;
	double tmp;
	if (d <= -1.62e-230) {
		tmp = t_2 * (t_4 * t_3);
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.1e-88) {
		tmp = t_4 * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = t_2 * (t_3 * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) / d
	t_1 = l / t_0
	t_2 = math.sqrt((d / l))
	t_3 = math.sqrt((d / h))
	t_4 = (((h / (d / (M * D))) / t_1) * -0.125) + 1.0
	tmp = 0
	if d <= -1.62e-230:
		tmp = t_2 * (t_4 * t_3)
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 4.1e-88:
		tmp = t_4 * (d / math.pow((h * l), 0.5))
	else:
		tmp = t_2 * (t_3 * ((h * (-0.125 * (t_0 / t_1))) + 1.0))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = Float64(l / t_0)
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(d / h))
	t_4 = Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / t_1) * -0.125) + 1.0)
	tmp = 0.0
	if (d <= -1.62e-230)
		tmp = Float64(t_2 * Float64(t_4 * t_3));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 4.1e-88)
		tmp = Float64(t_4 * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = Float64(t_2 * Float64(t_3 * Float64(Float64(h * Float64(-0.125 * Float64(t_0 / t_1))) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) / d;
	t_1 = l / t_0;
	t_2 = sqrt((d / l));
	t_3 = sqrt((d / h));
	t_4 = (((h / (d / (M * D))) / t_1) * -0.125) + 1.0;
	tmp = 0.0;
	if (d <= -1.62e-230)
		tmp = t_2 * (t_4 * t_3);
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 4.1e-88)
		tmp = t_4 * (d / ((h * l) ^ 0.5));
	else
		tmp = t_2 * (t_3 * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(l / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.62e-230], N[(t$95$2 * N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e-88], N[(t$95$4 * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$3 * N[(N[(h * N[(-0.125 * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.62000000000000002e-230

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{M \cdot D}{d}\right) \cdot \frac{1}{4}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{M \cdot D}{d}\right) \cdot \frac{1}{4}}{\frac{\ell}{\frac{M \cdot D}{d}} \cdot \color{blue}{-2}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \color{blue}{\frac{\frac{1}{4}}{-2}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \left(-1 \cdot \color{blue}{\frac{1}{8}}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \left(-1 \cdot {\frac{1}{2}}^{\color{blue}{3}}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \left(-1 \cdot {\left(\frac{1}{2}\right)}^{3}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}}\right), \color{blue}{\left(-1 \cdot {\left(\frac{1}{2}\right)}^{3}\right)}\right)\right)\right)\right) \]

   6. Applied egg-rr 80.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125}\right)\right) \]

   if -1.62000000000000002e-230 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 30.6%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 4.1000000000000001e-88

   1. Initial program 35.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]

   if 4.1000000000000001e-88 < d

   1. Initial program 79.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(h \cdot \color{blue}{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right), \color{blue}{h}\right)\right)\right)\right) \]

   6. Applied egg-rr 84.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125\right) \cdot h}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.62 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}}\right) + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \frac{\ell}{t\_0}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{t\_0}{t\_1}\right) + 1\right)\right)\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{t\_1} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1 (/ l t_0))
        (t_2
         (*
          (sqrt (/ d l))
          (* (sqrt (/ d h)) (+ (* h (* -0.125 (/ t_0 t_1))) 1.0)))))
   (if (<= d -1.3e-229)
     t_2
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (if (<= d 4.2e-88)
         (*
          (+ (* (/ (/ h (/ d (* M D))) t_1) -0.125) 1.0)
          (/ d (pow (* h l) 0.5)))
         t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = l / t_0;
	double t_2 = sqrt((d / l)) * (sqrt((d / h)) * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	double tmp;
	if (d <= -1.3e-229) {
		tmp = t_2;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.2e-88) {
		tmp = ((((h / (d / (M * D))) / t_1) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (m * d_1) / d
    t_1 = l / t_0
    t_2 = sqrt((d / l)) * (sqrt((d / h)) * ((h * ((-0.125d0) * (t_0 / t_1))) + 1.0d0))
    if (d <= (-1.3d-229)) then
        tmp = t_2
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else if (d <= 4.2d-88) then
        tmp = ((((h / (d / (m * d_1))) / t_1) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = l / t_0;
	double t_2 = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	double tmp;
	if (d <= -1.3e-229) {
		tmp = t_2;
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else if (d <= 4.2e-88) {
		tmp = ((((h / (d / (M * D))) / t_1) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (M * D) / d
	t_1 = l / t_0
	t_2 = math.sqrt((d / l)) * (math.sqrt((d / h)) * ((h * (-0.125 * (t_0 / t_1))) + 1.0))
	tmp = 0
	if d <= -1.3e-229:
		tmp = t_2
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	elif d <= 4.2e-88:
		tmp = ((((h / (d / (M * D))) / t_1) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = Float64(l / t_0)
	t_2 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(Float64(h * Float64(-0.125 * Float64(t_0 / t_1))) + 1.0)))
	tmp = 0.0
	if (d <= -1.3e-229)
		tmp = t_2;
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	elseif (d <= 4.2e-88)
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / t_1) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * D) / d;
	t_1 = l / t_0;
	t_2 = sqrt((d / l)) * (sqrt((d / h)) * ((h * (-0.125 * (t_0 / t_1))) + 1.0));
	tmp = 0.0;
	if (d <= -1.3e-229)
		tmp = t_2;
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	elseif (d <= 4.2e-88)
		tmp = ((((h / (d / (M * D))) / t_1) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(l / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(-0.125 * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e-229], t$95$2, If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-88], N[(N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.3000000000000001e-229 or 4.1999999999999999e-88 < d

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 81.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(h \cdot \color{blue}{\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{h}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right), \color{blue}{h}\right)\right)\right)\right) \]

   6. Applied egg-rr 80.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125\right) \cdot h}\right)\right) \]

   if -1.3000000000000001e-229 < d < -3.19999999999999978e-302

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 35.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 30.6%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 41.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 41.1%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d < 4.1999999999999999e-88

   1. Initial program 35.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}}\right) + 1\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{\frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{d}}}\right) + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.4e+215)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= l -1.65e-11)
     (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
     (if (<= l -2e-311)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (*
        (+ (* (/ (/ h (/ d (* M D))) (/ l (/ (* M D) d))) -0.125) 1.0)
        (/ d (pow (* h l) 0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e+215) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (l <= -1.65e-11) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-6.4d+215)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (l <= (-1.65d-11)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= (-2d-311)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else
        tmp = ((((h / (d / (m * d_1))) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e+215) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -1.65e-11) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.4e+215:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif l <= -1.65e-11:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= -2e-311:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	else:
		tmp = ((((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.4e+215)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (l <= -1.65e-11)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -2e-311)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / Float64(d / Float64(M * D))) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.4e+215)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (l <= -1.65e-11)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= -2e-311)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	else
		tmp = ((((h / (d / (M * D))) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.4e+215], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -1.65e-11], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.3999999999999997e215

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 16.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 16.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 16.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 69.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 69.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -6.3999999999999997e215 < l < -1.6500000000000001e-11

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -1.6500000000000001e-11 < l < -1.9999999999999e-311

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 78.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 49.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 49.7%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -1.9999999999999e-311 < l

   1. Initial program 63.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{h \cdot -0.125}{\frac{\frac{\frac{d}{M}}{D}}{\frac{\frac{D}{\frac{d}{M}}}{\ell}}} + 1\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9.5e+219)
   (sqrt (* (/ d l) (/ d h)))
   (if (<= l -2.1e-11)
     (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
     (if (<= l -2e-311)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (*
        (+ (/ (* h -0.125) (/ (/ (/ d M) D) (/ (/ D (/ d M)) l))) 1.0)
        (* d (pow (* h l) -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.5e+219) {
		tmp = sqrt(((d / l) * (d / h)));
	} else if (l <= -2.1e-11) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = (((h * -0.125) / (((d / M) / D) / ((D / (d / M)) / l))) + 1.0) * (d * pow((h * l), -0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-9.5d+219)) then
        tmp = sqrt(((d / l) * (d / h)))
    else if (l <= (-2.1d-11)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= (-2d-311)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else
        tmp = (((h * (-0.125d0)) / (((d / m) / d_1) / ((d_1 / (d / m)) / l))) + 1.0d0) * (d * ((h * l) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.5e+219) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -2.1e-11) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = (((h * -0.125) / (((d / M) / D) / ((D / (d / M)) / l))) + 1.0) * (d * Math.pow((h * l), -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -9.5e+219:
		tmp = math.sqrt(((d / l) * (d / h)))
	elif l <= -2.1e-11:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= -2e-311:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	else:
		tmp = (((h * -0.125) / (((d / M) / D) / ((D / (d / M)) / l))) + 1.0) * (d * math.pow((h * l), -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9.5e+219)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	elseif (l <= -2.1e-11)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -2e-311)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	else
		tmp = Float64(Float64(Float64(Float64(h * -0.125) / Float64(Float64(Float64(d / M) / D) / Float64(Float64(D / Float64(d / M)) / l))) + 1.0) * Float64(d * (Float64(h * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -9.5e+219)
		tmp = sqrt(((d / l) * (d / h)));
	elseif (l <= -2.1e-11)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= -2e-311)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	else
		tmp = (((h * -0.125) / (((d / M) / D) / ((D / (d / M)) / l))) + 1.0) * (d * ((h * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9.5e+219], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -2.1e-11], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * -0.125), $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision] / N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.49999999999999959e219

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 16.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 16.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 16.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 69.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 69.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -9.49999999999999959e219 < l < -2.0999999999999999e-11

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -2.0999999999999999e-11 < l < -1.9999999999999e-311

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 78.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 44.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 49.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 49.7%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -1.9999999999999e-311 < l

   1. Initial program 63.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 68.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 68.3%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{\left(\frac{-0.125 \cdot h}{\frac{\frac{\frac{d}{M}}{D}}{\frac{\frac{D}{\frac{d}{M}}}{\ell}}} + 1\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{h \cdot -0.125}{\frac{\frac{\frac{d}{M}}{D}}{\frac{\frac{D}{\frac{d}{M}}}{\ell}}} + 1\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_2 := D \cdot \left(D \cdot \left(M \cdot M\right)\right)\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{t\_2 \cdot \left(t\_0 \cdot 0.125\right)}{d}\\ \mathbf{elif}\;\ell \leq 82000000:\\ \;\;\;\;\frac{t\_2 \cdot \left(-0.125 \cdot t\_0\right)}{d}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* l (* l l)))))
        (t_1 (sqrt (* (/ d l) (/ d h))))
        (t_2 (* D (* D (* M M)))))
   (if (<= l -3.2e+218)
     t_1
     (if (<= l -1.7e-13)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.3e-308)
         (/ (* t_2 (* t_0 0.125)) d)
         (if (<= l 82000000.0)
           (/ (* t_2 (* -0.125 t_0)) d)
           (if (<= l 3.2e+169) (* d (sqrt (/ (/ 1.0 l) h))) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / (l * (l * l))));
	double t_1 = sqrt(((d / l) * (d / h)));
	double t_2 = D * (D * (M * M));
	double tmp;
	if (l <= -3.2e+218) {
		tmp = t_1;
	} else if (l <= -1.7e-13) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = (t_2 * (t_0 * 0.125)) / d;
	} else if (l <= 82000000.0) {
		tmp = (t_2 * (-0.125 * t_0)) / d;
	} else if (l <= 3.2e+169) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((h / (l * (l * l))))
    t_1 = sqrt(((d / l) * (d / h)))
    t_2 = d_1 * (d_1 * (m * m))
    if (l <= (-3.2d+218)) then
        tmp = t_1
    else if (l <= (-1.7d-13)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.3d-308) then
        tmp = (t_2 * (t_0 * 0.125d0)) / d
    else if (l <= 82000000.0d0) then
        tmp = (t_2 * ((-0.125d0) * t_0)) / d
    else if (l <= 3.2d+169) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / (l * (l * l))));
	double t_1 = Math.sqrt(((d / l) * (d / h)));
	double t_2 = D * (D * (M * M));
	double tmp;
	if (l <= -3.2e+218) {
		tmp = t_1;
	} else if (l <= -1.7e-13) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = (t_2 * (t_0 * 0.125)) / d;
	} else if (l <= 82000000.0) {
		tmp = (t_2 * (-0.125 * t_0)) / d;
	} else if (l <= 3.2e+169) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / (l * (l * l))))
	t_1 = math.sqrt(((d / l) * (d / h)))
	t_2 = D * (D * (M * M))
	tmp = 0
	if l <= -3.2e+218:
		tmp = t_1
	elif l <= -1.7e-13:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.3e-308:
		tmp = (t_2 * (t_0 * 0.125)) / d
	elif l <= 82000000.0:
		tmp = (t_2 * (-0.125 * t_0)) / d
	elif l <= 3.2e+169:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	t_1 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_2 = Float64(D * Float64(D * Float64(M * M)))
	tmp = 0.0
	if (l <= -3.2e+218)
		tmp = t_1;
	elseif (l <= -1.7e-13)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.3e-308)
		tmp = Float64(Float64(t_2 * Float64(t_0 * 0.125)) / d);
	elseif (l <= 82000000.0)
		tmp = Float64(Float64(t_2 * Float64(-0.125 * t_0)) / d);
	elseif (l <= 3.2e+169)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l * (l * l))));
	t_1 = sqrt(((d / l) * (d / h)));
	t_2 = D * (D * (M * M));
	tmp = 0.0;
	if (l <= -3.2e+218)
		tmp = t_1;
	elseif (l <= -1.7e-13)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.3e-308)
		tmp = (t_2 * (t_0 * 0.125)) / d;
	elseif (l <= 82000000.0)
		tmp = (t_2 * (-0.125 * t_0)) / d;
	elseif (l <= 3.2e+169)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.2e+218], t$95$1, If[LessEqual[l, -1.7e-13], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-308], N[(N[(t$95$2 * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[l, 82000000.0], N[(N[(t$95$2 * N[(-0.125 * t$95$0), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[l, 3.2e+169], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -3.19999999999999987e218 or 3.1999999999999998e169 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 24.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -3.19999999999999987e218 < l < -1.70000000000000008e-13

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -1.70000000000000008e-13 < l < 2.2999999999999999e-308

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 77.2%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 45.7%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{-1}{8} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{d}\right) \]

   11. Applied egg-rr 2.2%

       \[\leadsto \color{blue}{\frac{\left(-\left(\frac{{\left(\frac{h}{\ell}\right)}^{0.5}}{\ell} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d}} \]

   12. Taylor expanded in h around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}, d\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{1}{8}\right), d\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(D \cdot D\right) \cdot {M}^{2}\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot \left(D \cdot {M}^{2}\right)\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot {M}^{2}\right)\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2}\right)\right)\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left(M \cdot M\right)\right)\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{1}{8}\right)\right), d\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \frac{1}{8}\right)\right), d\right) \]

       11. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \frac{1}{8}\right)\right), d\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

       17. *-lowering-*.f64 48.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \frac{1}{8}\right)\right), d\right) \]

   14. Simplified 48.8%

       \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot 0.125\right)}}{d} \]

   if 2.2999999999999999e-308 < l < 8.2e7

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 2.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{-1}{8} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{d}\right) \]

   11. Applied egg-rr 2.8%

       \[\leadsto \color{blue}{\frac{\left(-\left(\frac{{\left(\frac{h}{\ell}\right)}^{0.5}}{\ell} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d}} \]

   12. Taylor expanded in l around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}, d\right) \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       5. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(-1 \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left({M}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right), d\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right), d\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-1}{8}\right)\right), d\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-1}{8}\right)\right), d\right) \]

   14. Simplified 49.5%

       \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot -0.125\right)}}{d} \]

   if 8.2e7 < l < 3.1999999999999998e169

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 59.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 60.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 60.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot 0.125\right)}{d}\\ \mathbf{elif}\;\ell \leq 82000000:\\ \;\;\;\;\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(M \cdot M\right)\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_2 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;t\_2 \cdot \left(0.125 \cdot \left(D \cdot \frac{t\_0}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;\frac{\left(D \cdot t\_0\right) \cdot \left(-0.125 \cdot t\_2\right)}{d}\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{+168}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* D (* M M)))
        (t_1 (sqrt (* (/ d l) (/ d h))))
        (t_2 (sqrt (/ h (* l (* l l))))))
   (if (<= l -2.25e+220)
     t_1
     (if (<= l -4.2e-11)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.3e-308)
         (* t_2 (* 0.125 (* D (/ t_0 d))))
         (if (<= l 480000000.0)
           (/ (* (* D t_0) (* -0.125 t_2)) d)
           (if (<= l 1.66e+168) (* d (sqrt (/ (/ 1.0 l) h))) t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M * M);
	double t_1 = sqrt(((d / l) * (d / h)));
	double t_2 = sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -2.25e+220) {
		tmp = t_1;
	} else if (l <= -4.2e-11) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_2 * (0.125 * (D * (t_0 / d)));
	} else if (l <= 480000000.0) {
		tmp = ((D * t_0) * (-0.125 * t_2)) / d;
	} else if (l <= 1.66e+168) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * (m * m)
    t_1 = sqrt(((d / l) * (d / h)))
    t_2 = sqrt((h / (l * (l * l))))
    if (l <= (-2.25d+220)) then
        tmp = t_1
    else if (l <= (-4.2d-11)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.3d-308) then
        tmp = t_2 * (0.125d0 * (d_1 * (t_0 / d)))
    else if (l <= 480000000.0d0) then
        tmp = ((d_1 * t_0) * ((-0.125d0) * t_2)) / d
    else if (l <= 1.66d+168) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M * M);
	double t_1 = Math.sqrt(((d / l) * (d / h)));
	double t_2 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -2.25e+220) {
		tmp = t_1;
	} else if (l <= -4.2e-11) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_2 * (0.125 * (D * (t_0 / d)));
	} else if (l <= 480000000.0) {
		tmp = ((D * t_0) * (-0.125 * t_2)) / d;
	} else if (l <= 1.66e+168) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = D * (M * M)
	t_1 = math.sqrt(((d / l) * (d / h)))
	t_2 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if l <= -2.25e+220:
		tmp = t_1
	elif l <= -4.2e-11:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.3e-308:
		tmp = t_2 * (0.125 * (D * (t_0 / d)))
	elif l <= 480000000.0:
		tmp = ((D * t_0) * (-0.125 * t_2)) / d
	elif l <= 1.66e+168:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M * M))
	t_1 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_2 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -2.25e+220)
		tmp = t_1;
	elseif (l <= -4.2e-11)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.3e-308)
		tmp = Float64(t_2 * Float64(0.125 * Float64(D * Float64(t_0 / d))));
	elseif (l <= 480000000.0)
		tmp = Float64(Float64(Float64(D * t_0) * Float64(-0.125 * t_2)) / d);
	elseif (l <= 1.66e+168)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = D * (M * M);
	t_1 = sqrt(((d / l) * (d / h)));
	t_2 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -2.25e+220)
		tmp = t_1;
	elseif (l <= -4.2e-11)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.3e-308)
		tmp = t_2 * (0.125 * (D * (t_0 / d)));
	elseif (l <= 480000000.0)
		tmp = ((D * t_0) * (-0.125 * t_2)) / d;
	elseif (l <= 1.66e+168)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.25e+220], t$95$1, If[LessEqual[l, -4.2e-11], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-308], N[(t$95$2 * N[(0.125 * N[(D * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 480000000.0], N[(N[(N[(D * t$95$0), $MachinePrecision] * N[(-0.125 * t$95$2), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[l, 1.66e+168], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -2.25000000000000005e220 or 1.6600000000000001e168 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 24.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -2.25000000000000005e220 < l < -4.1999999999999997e-11

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -4.1999999999999997e-11 < l < 2.2999999999999999e-308

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 78.9%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{-1 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(-1 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       7. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       10. metadata-eval N/A

           \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{\color{blue}{h}}{{\ell}^{3}}} \]

       11. *-commutative N/A

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

   11. Simplified 47.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot 0.125\right)} \]

   if 2.2999999999999999e-308 < l < 4.8e8

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 2.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{-1}{8} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{d}\right) \]

   11. Applied egg-rr 2.8%

       \[\leadsto \color{blue}{\frac{\left(-\left(\frac{{\left(\frac{h}{\ell}\right)}^{0.5}}{\ell} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d}} \]

   12. Taylor expanded in l around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}, d\right) \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       5. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(-1 \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \left(\mathsf{neg}\left({M}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), d\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right), d\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right), d\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-1}{8}\right)\right), d\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({D}^{2} \cdot {M}^{2}\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{-1}{8}\right)\right), d\right) \]

   14. Simplified 49.5%

       \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot -0.125\right)}}{d} \]

   if 4.8e8 < l < 1.6600000000000001e168

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 59.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 60.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 60.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{+168}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;t\_1 \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 17000000:\\ \;\;\;\;\frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(t\_1 \cdot \left(M \cdot M\right)\right)}{d}\\ \mathbf{elif}\;\ell \leq 1.96 \cdot 10^{+170}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))) (t_1 (sqrt (/ h (* l (* l l))))))
   (if (<= l -1.25e+219)
     t_0
     (if (<= l -1.95e-13)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.3e-308)
         (* t_1 (* 0.125 (* D (/ (* D (* M M)) d))))
         (if (<= l 17000000.0)
           (/ (* (* -0.125 (* D D)) (* t_1 (* M M))) d)
           (if (<= l 1.96e+170) (* d (sqrt (/ (/ 1.0 l) h))) t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -1.25e+219) {
		tmp = t_0;
	} else if (l <= -1.95e-13) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 17000000.0) {
		tmp = ((-0.125 * (D * D)) * (t_1 * (M * M))) / d;
	} else if (l <= 1.96e+170) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = sqrt((h / (l * (l * l))))
    if (l <= (-1.25d+219)) then
        tmp = t_0
    else if (l <= (-1.95d-13)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.3d-308) then
        tmp = t_1 * (0.125d0 * (d_1 * ((d_1 * (m * m)) / d)))
    else if (l <= 17000000.0d0) then
        tmp = (((-0.125d0) * (d_1 * d_1)) * (t_1 * (m * m))) / d
    else if (l <= 1.96d+170) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double t_1 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -1.25e+219) {
		tmp = t_0;
	} else if (l <= -1.95e-13) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 17000000.0) {
		tmp = ((-0.125 * (D * D)) * (t_1 * (M * M))) / d;
	} else if (l <= 1.96e+170) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if l <= -1.25e+219:
		tmp = t_0
	elif l <= -1.95e-13:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.3e-308:
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)))
	elif l <= 17000000.0:
		tmp = ((-0.125 * (D * D)) * (t_1 * (M * M))) / d
	elif l <= 1.96e+170:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -1.25e+219)
		tmp = t_0;
	elseif (l <= -1.95e-13)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.3e-308)
		tmp = Float64(t_1 * Float64(0.125 * Float64(D * Float64(Float64(D * Float64(M * M)) / d))));
	elseif (l <= 17000000.0)
		tmp = Float64(Float64(Float64(-0.125 * Float64(D * D)) * Float64(t_1 * Float64(M * M))) / d);
	elseif (l <= 1.96e+170)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -1.25e+219)
		tmp = t_0;
	elseif (l <= -1.95e-13)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.3e-308)
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	elseif (l <= 17000000.0)
		tmp = ((-0.125 * (D * D)) * (t_1 * (M * M))) / d;
	elseif (l <= 1.96e+170)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.25e+219], t$95$0, If[LessEqual[l, -1.95e-13], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-308], N[(t$95$1 * N[(0.125 * N[(D * N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 17000000.0], N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[l, 1.96e+170], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.25e219 or 1.96000000000000011e170 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 24.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -1.25e219 < l < -1.95000000000000002e-13

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -1.95000000000000002e-13 < l < 2.2999999999999999e-308

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 78.9%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{-1 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(-1 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       7. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       10. metadata-eval N/A

           \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{\color{blue}{h}}{{\ell}^{3}}} \]

       11. *-commutative N/A

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

   11. Simplified 47.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot 0.125\right)} \]

   if 2.2999999999999999e-308 < l < 1.7e7

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\color{blue}{d}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{\color{blue}{d}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right), \color{blue}{d}\right) \]

   9. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}} \]

   if 1.7e7 < l < 1.96000000000000011e170

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 59.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 60.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 60.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 17000000:\\ \;\;\;\;\frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(M \cdot M\right)\right)}{d}\\ \mathbf{elif}\;\ell \leq 1.96 \cdot 10^{+170}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;t\_1 \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 15600000:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot \left(t\_1 \cdot \left(D \cdot D\right)\right)\right) \cdot \frac{-0.125}{d}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))) (t_1 (sqrt (/ h (* l (* l l))))))
   (if (<= l -2.6e+218)
     t_0
     (if (<= l -3.5e-12)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.3e-308)
         (* t_1 (* 0.125 (* D (/ (* D (* M M)) d))))
         (if (<= l 15600000.0)
           (* (* (* M M) (* t_1 (* D D))) (/ -0.125 d))
           (if (<= l 1.65e+169) (* d (sqrt (/ (/ 1.0 l) h))) t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -2.6e+218) {
		tmp = t_0;
	} else if (l <= -3.5e-12) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 15600000.0) {
		tmp = ((M * M) * (t_1 * (D * D))) * (-0.125 / d);
	} else if (l <= 1.65e+169) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = sqrt((h / (l * (l * l))))
    if (l <= (-2.6d+218)) then
        tmp = t_0
    else if (l <= (-3.5d-12)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.3d-308) then
        tmp = t_1 * (0.125d0 * (d_1 * ((d_1 * (m * m)) / d)))
    else if (l <= 15600000.0d0) then
        tmp = ((m * m) * (t_1 * (d_1 * d_1))) * ((-0.125d0) / d)
    else if (l <= 1.65d+169) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double t_1 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -2.6e+218) {
		tmp = t_0;
	} else if (l <= -3.5e-12) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 15600000.0) {
		tmp = ((M * M) * (t_1 * (D * D))) * (-0.125 / d);
	} else if (l <= 1.65e+169) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if l <= -2.6e+218:
		tmp = t_0
	elif l <= -3.5e-12:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.3e-308:
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)))
	elif l <= 15600000.0:
		tmp = ((M * M) * (t_1 * (D * D))) * (-0.125 / d)
	elif l <= 1.65e+169:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -2.6e+218)
		tmp = t_0;
	elseif (l <= -3.5e-12)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.3e-308)
		tmp = Float64(t_1 * Float64(0.125 * Float64(D * Float64(Float64(D * Float64(M * M)) / d))));
	elseif (l <= 15600000.0)
		tmp = Float64(Float64(Float64(M * M) * Float64(t_1 * Float64(D * D))) * Float64(-0.125 / d));
	elseif (l <= 1.65e+169)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -2.6e+218)
		tmp = t_0;
	elseif (l <= -3.5e-12)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.3e-308)
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	elseif (l <= 15600000.0)
		tmp = ((M * M) * (t_1 * (D * D))) * (-0.125 / d);
	elseif (l <= 1.65e+169)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.6e+218], t$95$0, If[LessEqual[l, -3.5e-12], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-308], N[(t$95$1 * N[(0.125 * N[(D * N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 15600000.0], N[(N[(N[(M * M), $MachinePrecision] * N[(t$95$1 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+169], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -2.60000000000000002e218 or 1.6499999999999998e169 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 24.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -2.60000000000000002e218 < l < -3.5e-12

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -3.5e-12 < l < 2.2999999999999999e-308

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 78.9%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{-1 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(-1 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       7. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       10. metadata-eval N/A

           \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{\color{blue}{h}}{{\ell}^{3}}} \]

       11. *-commutative N/A

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

   11. Simplified 47.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot 0.125\right)} \]

   if 2.2999999999999999e-308 < l < 1.56e7

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\color{blue}{d}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{\color{blue}{d}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{d} \]

      4. associate-/l* N/A

         \[\leadsto \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{\frac{-1}{8}}{d}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{\frac{-1}{8}}{d}\right)}\right) \]

   7. Simplified 49.5%

      \[\leadsto \color{blue}{\left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\right) \cdot \frac{-0.125}{d}} \]

   if 1.56e7 < l < 1.6499999999999998e169

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 59.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 60.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 60.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 15600000:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right) \cdot \frac{-0.125}{d}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;t\_1 \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 47000000:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(t\_1 \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))) (t_1 (sqrt (/ h (* l (* l l))))))
   (if (<= l -4.9e+216)
     t_0
     (if (<= l -3e-11)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.3e-308)
         (* t_1 (* 0.125 (* D (/ (* D (* M M)) d))))
         (if (<= l 47000000.0)
           (* (* D D) (* t_1 (/ (* -0.125 (* M M)) d)))
           (if (<= l 1.25e+169) (* d (sqrt (/ (/ 1.0 l) h))) t_0)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double t_1 = sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -4.9e+216) {
		tmp = t_0;
	} else if (l <= -3e-11) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 47000000.0) {
		tmp = (D * D) * (t_1 * ((-0.125 * (M * M)) / d));
	} else if (l <= 1.25e+169) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    t_1 = sqrt((h / (l * (l * l))))
    if (l <= (-4.9d+216)) then
        tmp = t_0
    else if (l <= (-3d-11)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.3d-308) then
        tmp = t_1 * (0.125d0 * (d_1 * ((d_1 * (m * m)) / d)))
    else if (l <= 47000000.0d0) then
        tmp = (d_1 * d_1) * (t_1 * (((-0.125d0) * (m * m)) / d))
    else if (l <= 1.25d+169) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double t_1 = Math.sqrt((h / (l * (l * l))));
	double tmp;
	if (l <= -4.9e+216) {
		tmp = t_0;
	} else if (l <= -3e-11) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.3e-308) {
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 47000000.0) {
		tmp = (D * D) * (t_1 * ((-0.125 * (M * M)) / d));
	} else if (l <= 1.25e+169) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	t_1 = math.sqrt((h / (l * (l * l))))
	tmp = 0
	if l <= -4.9e+216:
		tmp = t_0
	elif l <= -3e-11:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.3e-308:
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)))
	elif l <= 47000000.0:
		tmp = (D * D) * (t_1 * ((-0.125 * (M * M)) / d))
	elif l <= 1.25e+169:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	t_1 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -4.9e+216)
		tmp = t_0;
	elseif (l <= -3e-11)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.3e-308)
		tmp = Float64(t_1 * Float64(0.125 * Float64(D * Float64(Float64(D * Float64(M * M)) / d))));
	elseif (l <= 47000000.0)
		tmp = Float64(Float64(D * D) * Float64(t_1 * Float64(Float64(-0.125 * Float64(M * M)) / d)));
	elseif (l <= 1.25e+169)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	t_1 = sqrt((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -4.9e+216)
		tmp = t_0;
	elseif (l <= -3e-11)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.3e-308)
		tmp = t_1 * (0.125 * (D * ((D * (M * M)) / d)));
	elseif (l <= 47000000.0)
		tmp = (D * D) * (t_1 * ((-0.125 * (M * M)) / d));
	elseif (l <= 1.25e+169)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.9e+216], t$95$0, If[LessEqual[l, -3e-11], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e-308], N[(t$95$1 * N[(0.125 * N[(D * N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 47000000.0], N[(N[(D * D), $MachinePrecision] * N[(t$95$1 * N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+169], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -4.90000000000000014e216 or 1.25000000000000004e169 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 24.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.3%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 57.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -4.90000000000000014e216 < l < -3e-11

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -3e-11 < l < 2.2999999999999999e-308

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 78.9%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{-1 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(-1 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       7. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       10. metadata-eval N/A

           \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{\color{blue}{h}}{{\ell}^{3}}} \]

       11. *-commutative N/A

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

   11. Simplified 47.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot 0.125\right)} \]

   if 2.2999999999999999e-308 < l < 4.7e7

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 73.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 73.1%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   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. *-commutative N/A

          \[\leadsto \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

       2. associate-/l* N/A

          \[\leadsto \left(\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

       3. associate-*l* N/A

          \[\leadsto \left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \frac{-1}{8} \]

       4. associate-*r* N/A

          \[\leadsto {D}^{2} \cdot \color{blue}{\left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

       5. *-commutative N/A

          \[\leadsto {D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)\right) \]

   11. Simplified 47.8%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)} \]

   if 4.7e7 < l < 1.25000000000000004e169

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 71.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 59.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 60.3%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \ell\right), h\right)\right)\right) \]

   9. Applied egg-rr 60.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 47000000:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{\frac{h \cdot 0.25}{t\_0}}{d \cdot -2} \cdot \frac{M \cdot D}{\ell} + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{t\_0}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D))))
   (if (<= d -3.5e-162)
     (*
      (+ (* (/ (/ (* h 0.25) t_0) (* d -2.0)) (/ (* M D) l)) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (*
        (+ (* (/ (/ h t_0) (/ l (/ (* M D) d))) -0.125) 1.0)
        (/ d (pow (* h l) 0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -3.5e-162) {
		tmp = (((((h * 0.25) / t_0) / (d * -2.0)) * ((M * D) / l)) + 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / (m * d_1)
    if (d <= (-3.5d-162)) then
        tmp = (((((h * 0.25d0) / t_0) / (d * (-2.0d0))) * ((m * d_1) / l)) + 1.0d0) * sqrt(((d / l) * (d / h)))
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else
        tmp = ((((h / t_0) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -3.5e-162) {
		tmp = (((((h * 0.25) / t_0) / (d * -2.0)) * ((M * D) / l)) + 1.0) * Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	tmp = 0
	if d <= -3.5e-162:
		tmp = (((((h * 0.25) / t_0) / (d * -2.0)) * ((M * D) / l)) + 1.0) * math.sqrt(((d / l) * (d / h)))
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	else:
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	tmp = 0.0
	if (d <= -3.5e-162)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(h * 0.25) / t_0) / Float64(d * -2.0)) * Float64(Float64(M * D) / l)) + 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / t_0) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	tmp = 0.0;
	if (d <= -3.5e-162)
		tmp = (((((h * 0.25) / t_0) / (d * -2.0)) * ((M * D) / l)) + 1.0) * sqrt(((d / l) * (d / h)));
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	else
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.5e-162], N[(N[(N[(N[(N[(N[(h * 0.25), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.4999999999999999e-162

   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 82.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 81.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   8. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{h \cdot 0.25}{\frac{d}{M \cdot D}}}{d \cdot -2} \cdot \frac{M \cdot D}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -3.4999999999999999e-162 < d < -3.19999999999999978e-302

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 48.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 35.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 43.9%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{\frac{h \cdot 0.25}{\frac{d}{M \cdot D}}}{d \cdot -2} \cdot \frac{M \cdot D}{\ell} + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 69.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{-161}:\\ \;\;\;\;\left(\frac{\frac{h \cdot -0.125}{t\_0}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{t\_0}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D))))
   (if (<= d -2.5e-161)
     (*
      (+ (/ (/ (* h -0.125) t_0) (/ (/ l M) (/ D d))) 1.0)
      (sqrt (* (/ d l) (/ d h))))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (*
        (+ (* (/ (/ h t_0) (/ l (/ (* M D) d))) -0.125) 1.0)
        (/ d (pow (* h l) 0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -2.5e-161) {
		tmp = ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * sqrt(((d / l) * (d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / (m * d_1)
    if (d <= (-2.5d-161)) then
        tmp = ((((h * (-0.125d0)) / t_0) / ((l / m) / (d_1 / d))) + 1.0d0) * sqrt(((d / l) * (d / h)))
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else
        tmp = ((((h / t_0) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -2.5e-161) {
		tmp = ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	tmp = 0
	if d <= -2.5e-161:
		tmp = ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * math.sqrt(((d / l) * (d / h)))
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	else:
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	tmp = 0.0
	if (d <= -2.5e-161)
		tmp = Float64(Float64(Float64(Float64(Float64(h * -0.125) / t_0) / Float64(Float64(l / M) / Float64(D / d))) + 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / t_0) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	tmp = 0.0;
	if (d <= -2.5e-161)
		tmp = ((((h * -0.125) / t_0) / ((l / M) / (D / d))) + 1.0) * sqrt(((d / l) * (d / h)));
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	else
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e-161], N[(N[(N[(N[(N[(h * -0.125), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(l / M), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5e-161

   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 82.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 82.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right)\right) \cdot \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \]

      2. pow1/2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{1}}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      6. clear-num N/A

         \[\leadsto \left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right) \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot \frac{-1}{8} + 1\right), \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)}\right) \]

   8. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{-0.125 \cdot h}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -2.5e-161 < d < -3.19999999999999978e-302

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 48.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 35.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 43.9%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-161}:\\ \;\;\;\;\left(\frac{\frac{h \cdot -0.125}{\frac{d}{M \cdot D}}}{\frac{\frac{\ell}{M}}{\frac{D}{d}}} + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 68.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{-197}:\\ \;\;\;\;\left(\frac{h \cdot -0.125}{\frac{t\_0}{\frac{D}{d \cdot \frac{\ell}{M}}}} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{t\_0}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* M D))))
   (if (<= d -4.5e-197)
     (*
      (+ (/ (* h -0.125) (/ t_0 (/ D (* d (/ l M))))) 1.0)
      (sqrt (/ (/ d l) (/ h d))))
     (if (<= d -3.2e-302)
       (*
        -0.125
        (* (* M (/ M (- 0.0 d))) (* (sqrt (/ h (* l (* l l)))) (* D D))))
       (*
        (+ (* (/ (/ h t_0) (/ l (/ (* M D) d))) -0.125) 1.0)
        (/ d (pow (* h l) 0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -4.5e-197) {
		tmp = (((h * -0.125) / (t_0 / (D / (d * (l / M))))) + 1.0) * sqrt(((d / l) / (h / d)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / pow((h * l), 0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / (m * d_1)
    if (d <= (-4.5d-197)) then
        tmp = (((h * (-0.125d0)) / (t_0 / (d_1 / (d * (l / m))))) + 1.0d0) * sqrt(((d / l) / (h / d)))
    else if (d <= (-3.2d-302)) then
        tmp = (-0.125d0) * ((m * (m / (0.0d0 - d))) * (sqrt((h / (l * (l * l)))) * (d_1 * d_1)))
    else
        tmp = ((((h / t_0) / (l / ((m * d_1) / d))) * (-0.125d0)) + 1.0d0) * (d / ((h * l) ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (M * D);
	double tmp;
	if (d <= -4.5e-197) {
		tmp = (((h * -0.125) / (t_0 / (D / (d * (l / M))))) + 1.0) * Math.sqrt(((d / l) / (h / d)));
	} else if (d <= -3.2e-302) {
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (Math.sqrt((h / (l * (l * l)))) * (D * D)));
	} else {
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / Math.pow((h * l), 0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / (M * D)
	tmp = 0
	if d <= -4.5e-197:
		tmp = (((h * -0.125) / (t_0 / (D / (d * (l / M))))) + 1.0) * math.sqrt(((d / l) / (h / d)))
	elif d <= -3.2e-302:
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (math.sqrt((h / (l * (l * l)))) * (D * D)))
	else:
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / math.pow((h * l), 0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / Float64(M * D))
	tmp = 0.0
	if (d <= -4.5e-197)
		tmp = Float64(Float64(Float64(Float64(h * -0.125) / Float64(t_0 / Float64(D / Float64(d * Float64(l / M))))) + 1.0) * sqrt(Float64(Float64(d / l) / Float64(h / d))));
	elseif (d <= -3.2e-302)
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M / Float64(0.0 - d))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(D * D))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(h / t_0) / Float64(l / Float64(Float64(M * D) / d))) * -0.125) + 1.0) * Float64(d / (Float64(h * l) ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / (M * D);
	tmp = 0.0;
	if (d <= -4.5e-197)
		tmp = (((h * -0.125) / (t_0 / (D / (d * (l / M))))) + 1.0) * sqrt(((d / l) / (h / d)));
	elseif (d <= -3.2e-302)
		tmp = -0.125 * ((M * (M / (0.0 - d))) * (sqrt((h / (l * (l * l)))) * (D * D)));
	else
		tmp = ((((h / t_0) / (l / ((M * D) / d))) * -0.125) + 1.0) * (d / ((h * l) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.5e-197], N[(N[(N[(N[(h * -0.125), $MachinePrecision] / N[(t$95$0 / N[(D / N[(d * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] / N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-302], N[(-0.125 * N[(N[(M * N[(M / N[(0.0 - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(h / t$95$0), $MachinePrecision] / N[(l / N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Power[N[(h * l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.5000000000000001e-197

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 82.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{\ell}{d}}\right)}^{\frac{1}{2}}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left({\left(\frac{\ell}{d}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      3. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{\ell}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, d\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{\ell}{d}\right)}^{\frac{-1}{2}}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{\ell}{d}\right)}^{\left(\frac{1}{-2}\right)}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{\ell}{d}\right), \left(\frac{1}{-2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{h}, d\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \left(\frac{1}{-2}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

      8. metadata-eval 81.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, d\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, d\right)\right)\right) \]

   8. Applied egg-rr 81.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\ell}{d}\right)}^{-0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}} \]

   10. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{\left(1 + \frac{h \cdot -0.125}{\frac{\frac{d}{M \cdot D}}{\frac{D}{\frac{\ell}{M} \cdot d}}}\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}} \]

   if -4.5000000000000001e-197 < d < -3.19999999999999978e-302

   1. Initial program 43.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 43.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 43.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 31.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\mathsf{neg}\left(M \cdot M\right)}{d}\right)\right)\right) \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{\left(\mathsf{neg}\left(M\right)\right) \cdot M}{d}\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\left(\mathsf{neg}\left(M\right)\right) \cdot \color{blue}{\frac{M}{d}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(\frac{M}{d}\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(0 - M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \left(\frac{\color{blue}{M}}{d}\right)\right)\right)\right) \]

       7. /-lowering-/.f64 37.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, M\right), \mathsf{/.f64}\left(M, \color{blue}{d}\right)\right)\right)\right) \]

   11. Applied egg-rr 37.9%

       \[\leadsto -0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\left(\left(0 - M\right) \cdot \frac{M}{d}\right)}\right) \]

   if -3.19999999999999978e-302 < d

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-197}:\\ \;\;\;\;\left(\frac{h \cdot -0.125}{\frac{\frac{d}{M \cdot D}}{\frac{D}{d \cdot \frac{\ell}{M}}}} + 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-302}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \frac{M}{0 - d}\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right) \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -2.7e+219)
     t_0
     (if (<= l -4.8e-13)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l -2e-311)
         (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D (/ (* D (* M M)) d))))
         (if (<= l 7.8e+174) (* d (pow (* h l) -0.5)) t_0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -2.7e+219) {
		tmp = t_0;
	} else if (l <= -4.8e-13) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 7.8e+174) {
		tmp = d * pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-2.7d+219)) then
        tmp = t_0
    else if (l <= (-4.8d-13)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= (-2d-311)) then
        tmp = sqrt((h / (l * (l * l)))) * (0.125d0 * (d_1 * ((d_1 * (m * m)) / d)))
    else if (l <= 7.8d+174) then
        tmp = d * ((h * l) ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -2.7e+219) {
		tmp = t_0;
	} else if (l <= -4.8e-13) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-311) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (0.125 * (D * ((D * (M * M)) / d)));
	} else if (l <= 7.8e+174) {
		tmp = d * Math.pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -2.7e+219:
		tmp = t_0
	elif l <= -4.8e-13:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= -2e-311:
		tmp = math.sqrt((h / (l * (l * l)))) * (0.125 * (D * ((D * (M * M)) / d)))
	elif l <= 7.8e+174:
		tmp = d * math.pow((h * l), -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -2.7e+219)
		tmp = t_0;
	elseif (l <= -4.8e-13)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D * Float64(Float64(D * Float64(M * M)) / d))));
	elseif (l <= 7.8e+174)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -2.7e+219)
		tmp = t_0;
	elseif (l <= -4.8e-13)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= -2e-311)
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D * ((D * (M * M)) / d)));
	elseif (l <= 7.8e+174)
		tmp = d * ((h * l) ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.7e+219], t$95$0, If[LessEqual[l, -4.8e-13], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D * N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+174], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.6999999999999999e219 or 7.79999999999999962e174 < l

   1. Initial program 57.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 22.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 22.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 22.7%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 56.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 56.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -2.6999999999999999e219 < l < -4.7999999999999997e-13

   1. Initial program 60.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 58.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -4.7999999999999997e-13 < l < -1.9999999999999e-311

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\sqrt{\frac{1}{\frac{d}{h}}}\right)\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{d}{h}}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{d}{h}}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{d}{h}}\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right), \frac{-1}{8}\right), 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right)\right)\right) \]

   8. Applied egg-rr 78.6%

      \[\leadsto \frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\color{blue}{\frac{1}{\sqrt{\frac{d}{h}}}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(\sqrt{-1} \cdot \sqrt{-1}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. rem-square-sqrt N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{-1 \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(-1 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       7. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

       10. metadata-eval N/A

           \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{\color{blue}{h}}{{\ell}^{3}}} \]

       11. *-commutative N/A

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)}\right) \]

   11. Simplified 46.7%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l < 7.79999999999999962e174

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 40.1%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 40.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      10. metadata-eval 40.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 41.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+212}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-278}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+171}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -3e+212)
     t_0
     (if (<= l 2.3e-278)
       (* (- 0.0 d) (sqrt (/ (/ 1.0 h) l)))
       (if (<= l 8e+171) (* d (pow (* h l) -0.5)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -3e+212) {
		tmp = t_0;
	} else if (l <= 2.3e-278) {
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	} else if (l <= 8e+171) {
		tmp = d * pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-3d+212)) then
        tmp = t_0
    else if (l <= 2.3d-278) then
        tmp = (0.0d0 - d) * sqrt(((1.0d0 / h) / l))
    else if (l <= 8d+171) then
        tmp = d * ((h * l) ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -3e+212) {
		tmp = t_0;
	} else if (l <= 2.3e-278) {
		tmp = (0.0 - d) * Math.sqrt(((1.0 / h) / l));
	} else if (l <= 8e+171) {
		tmp = d * Math.pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -3e+212:
		tmp = t_0
	elif l <= 2.3e-278:
		tmp = (0.0 - d) * math.sqrt(((1.0 / h) / l))
	elif l <= 8e+171:
		tmp = d * math.pow((h * l), -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -3e+212)
		tmp = t_0;
	elseif (l <= 2.3e-278)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= 8e+171)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -3e+212)
		tmp = t_0;
	elseif (l <= 2.3e-278)
		tmp = (0.0 - d) * sqrt(((1.0 / h) / l));
	elseif (l <= 8e+171)
		tmp = d * ((h * l) ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3e+212], t$95$0, If[LessEqual[l, 2.3e-278], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+171], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3e212 or 7.99999999999999963e171 < l

   1. Initial program 58.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 22.4%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 22.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 22.4%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 55.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 55.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -3e212 < l < 2.30000000000000003e-278

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 74.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 9.0%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 9.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      10. metadata-eval 8.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

   10. Taylor expanded in l around -inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}, d\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right), d\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right), d\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right), d\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right), d\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - \sqrt{\frac{1}{h \cdot \ell}}\right), d\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right), d\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\left(\frac{1}{h \cdot \ell}\right)\right)\right), d\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{h}}{\ell}\right)\right)\right), d\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{h}\right), \ell\right)\right)\right), d\right) \]

       10. /-lowering-/.f64 42.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, h\right), \ell\right)\right)\right), d\right) \]

   12. Simplified 42.4%

       \[\leadsto \color{blue}{\left(0 - \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \cdot d \]

   if 2.30000000000000003e-278 < l < 7.99999999999999963e171

   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. 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 71.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 41.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      10. metadata-eval 42.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{-278}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+171}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 41.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-283}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+175}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -5.4e+217)
     t_0
     (if (<= l 2.6e-283)
       (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
       (if (<= l 2.2e+175) (* d (pow (* h l) -0.5)) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -5.4e+217) {
		tmp = t_0;
	} else if (l <= 2.6e-283) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 2.2e+175) {
		tmp = d * pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-5.4d+217)) then
        tmp = t_0
    else if (l <= 2.6d-283) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 2.2d+175) then
        tmp = d * ((h * l) ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -5.4e+217) {
		tmp = t_0;
	} else if (l <= 2.6e-283) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 2.2e+175) {
		tmp = d * Math.pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -5.4e+217:
		tmp = t_0
	elif l <= 2.6e-283:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 2.2e+175:
		tmp = d * math.pow((h * l), -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -5.4e+217)
		tmp = t_0;
	elseif (l <= 2.6e-283)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 2.2e+175)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -5.4e+217)
		tmp = t_0;
	elseif (l <= 2.6e-283)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 2.2e+175)
		tmp = d * ((h * l) ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.4e+217], t$95$0, If[LessEqual[l, 2.6e-283], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+175], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.40000000000000005e217 or 2.1999999999999999e175 < l

   1. Initial program 57.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 22.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 22.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 22.7%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 56.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 56.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -5.40000000000000005e217 < l < 2.6000000000000001e-283

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 75.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \sqrt{\frac{1}{\frac{h}{d}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)}{\color{blue}{\sqrt{\frac{h}{d}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right), \color{blue}{\left(\sqrt{\frac{h}{d}}\right)}\right) \]

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\frac{\frac{h}{\frac{d}{M \cdot D}}}{\frac{\ell}{\frac{M \cdot D}{d}}} \cdot -0.125 + 1\right)}{\sqrt{\frac{h}{d}}}} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \color{blue}{d}\right) \]

      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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(h \cdot \ell\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\ell \cdot h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(-1 \cdot d\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \left(\mathsf{neg}\left(d\right)\right)\right) \]

      11. neg-lowering-neg.f64 42.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, h\right)\right)\right), \mathsf{neg.f64}\left(d\right)\right) \]

   9. Simplified 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 2.6000000000000001e-283 < l < 2.1999999999999999e175

   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. 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 71.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 41.5%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      10. metadata-eval 42.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-283}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+175}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 39.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\ell}\right)\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 1.75e-90)
   (sqrt (* (/ d l) (/ d h)))
   (/ (* -0.125 (* (* M M) (* (* D D) (/ (sqrt (/ h l)) l)))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.75e-90) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = (-0.125 * ((M * M) * ((D * D) * (sqrt((h / l)) / l)))) / d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 1.75d-90) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = ((-0.125d0) * ((m * m) * ((d_1 * d_1) * (sqrt((h / l)) / l)))) / d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.75e-90) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = (-0.125 * ((M * M) * ((D * D) * (Math.sqrt((h / l)) / l)))) / d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 1.75e-90:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = (-0.125 * ((M * M) * ((D * D) * (math.sqrt((h / l)) / l)))) / d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 1.75e-90)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64(M * M) * Float64(Float64(D * D) * Float64(sqrt(Float64(h / l)) / l)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 1.75e-90)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = (-0.125 * ((M * M) * ((D * D) * (sqrt((h / l)) / l)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 1.75e-90], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(-0.125 * N[(N[(M * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.7499999999999999e-90

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 70.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 23.9%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 23.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 24.1%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 37.9%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 37.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if 1.7499999999999999e-90 < M

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*l* N/A

         \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right), \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{\ell}}\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{\ell}\right)\right), \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left({\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{d}{h}}\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{d}{h}\right)\right), \left(\color{blue}{1} - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\left(h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)}{\color{blue}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)}{\frac{\ell}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\color{blue}{\ell}}{\frac{\frac{M \cdot D}{d}}{-2}}}\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\color{blue}{-2 \cdot d}}}}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\left(\mathsf{neg}\left(2\right)\right) \cdot d}}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{M \cdot D}{\mathsf{neg}\left(2 \cdot d\right)}}}\right)\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{1}{4}\right) \cdot \frac{M \cdot D}{d}}{\frac{\ell}{M \cdot D} \cdot \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}}\right)\right)\right)\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}} \cdot \color{blue}{\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{1}{4}}{\frac{\ell}{M \cdot D}}\right), \color{blue}{\left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(h \cdot \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\color{blue}{\frac{M \cdot D}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \left(\frac{\ell}{M \cdot D}\right)\right), \left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \left(M \cdot D\right)\right)\right), \left(\frac{\frac{M \cdot D}{\color{blue}{d}}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \left(\frac{\frac{M \cdot D}{d}}{\mathsf{neg}\left(2 \cdot d\right)}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot d\right)\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot d}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(\color{blue}{2} \cdot d\right)\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(d, h\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \frac{1}{4}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(M, D\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), \left(\mathsf{neg}\left(d \cdot 2\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 65.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot 0.25}{\frac{\ell}{M \cdot D}} \cdot \frac{\frac{M \cdot D}{d}}{d \cdot -2}}\right)\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right), \color{blue}{\left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{h}{{\ell}^{3}}}\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{h}{{\ell}^{3}}\right)\right), \left({D}^{2}\right)\right), \left(\frac{\color{blue}{{M}^{2}} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left({\ell}^{3}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{\color{blue}{M}}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \left(\ell \cdot {\ell}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left({\ell}^{2}\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{d}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right), \color{blue}{d}\right)\right)\right) \]

   9. Simplified 20.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)}{d} \cdot \frac{-1}{8} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{\color{blue}{d}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot D\right)\right) \cdot \left(-1 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{d}\right) \]

   11. Applied egg-rr 3.4%

       \[\leadsto \color{blue}{\frac{\left(-\left(\frac{{\left(\frac{h}{\ell}\right)}^{0.5}}{\ell} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d}} \]

   12. Taylor expanded in l around -inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}, \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{h}{\ell}}\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{h}{\ell}}\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \sqrt{\frac{h}{\ell}}\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - \sqrt{\frac{h}{\ell}}\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt{\frac{h}{\ell}}\right)\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\left(\frac{h}{\ell}\right)\right)\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

       8. /-lowering-/.f64 45.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(h, \ell\right)\right)\right), \ell\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(M, M\right)\right)\right), \frac{-1}{8}\right), d\right) \]

   14. Simplified 45.9%

       \[\leadsto \frac{\left(-\left(\frac{\color{blue}{0 - \sqrt{\frac{h}{\ell}}}}{\ell} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell}}}{\ell}\right)\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 36.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-231}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+175}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d l) (/ d h)))))
   (if (<= l -5.5e-231)
     t_0
     (if (<= l 1.45e+175) (* d (pow (* h l) -0.5)) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -5.5e-231) {
		tmp = t_0;
	} else if (l <= 1.45e+175) {
		tmp = d * pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h)))
    if (l <= (-5.5d-231)) then
        tmp = t_0
    else if (l <= 1.45d+175) then
        tmp = d * ((h * l) ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h)));
	double tmp;
	if (l <= -5.5e-231) {
		tmp = t_0;
	} else if (l <= 1.45e+175) {
		tmp = d * Math.pow((h * l), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / l) * (d / h)))
	tmp = 0
	if l <= -5.5e-231:
		tmp = t_0
	elif l <= 1.45e+175:
		tmp = d * math.pow((h * l), -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (l <= -5.5e-231)
		tmp = t_0;
	elseif (l <= 1.45e+175)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (l <= -5.5e-231)
		tmp = t_0;
	elseif (l <= 1.45e+175)
		tmp = d * ((h * l) ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.5e-231], t$95$0, If[LessEqual[l, 1.45e+175], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.49999999999999951e-231 or 1.45e175 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 69.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 11.7%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 11.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. unpow1/2 N/A

         \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

      9. *-lowering-*.f64 11.1%

         \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

   9. Applied egg-rr 11.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

       6. sqrt-unprod N/A

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{d}{\ell} \cdot \frac{d}{h}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{d}{\ell}\right), \left(\frac{d}{h}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \left(\frac{d}{h}\right)\right)\right) \]

       10. /-lowering-/.f64 38.3%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(d, \ell\right), \mathsf{/.f64}\left(d, h\right)\right)\right) \]

   11. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -5.49999999999999951e-231 < l < 1.45e175

   1. Initial program 69.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 73.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
   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 37.8%

         \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

   7. Simplified 37.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

      10. metadata-eval 38.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

   9. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+175}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

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 * l) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 71.2%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
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 22.5%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

7. Simplified 22.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(h \cdot \ell\right)}^{\left(\frac{1}{-2}\right)}\right), d\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(h \cdot \ell\right), \left(\frac{1}{-2}\right)\right), d\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\ell \cdot h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \left(\frac{1}{-2}\right)\right), d\right) \]

   10. metadata-eval 22.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, h\right), \frac{-1}{2}\right), d\right) \]

9. Applied egg-rr 22.4%

   \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot d} \]

10. Final simplification 22.4%

    \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 28: 26.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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 / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 71.2%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{\frac{\frac{M \cdot D}{d}}{-2}}{\ell}\right)\right)} \]
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 22.5%

      \[\leadsto \mathsf{*.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(h, \ell\right)\right)\right)\right) \]

7. Simplified 22.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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. unpow1/2 N/A

      \[\leadsto d \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

   4. div-inv N/A

      \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left({\left(h \cdot \ell\right)}^{\frac{1}{2}}\right)}\right) \]

   6. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(d, \left(\sqrt{h \cdot \ell}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(h \cdot \ell\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\left(\ell \cdot h\right)\right)\right) \]

   9. *-lowering-*.f64 22.3%

      \[\leadsto \mathsf{/.f64}\left(d, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\ell, h\right)\right)\right) \]

9. Applied egg-rr 22.3%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

10. Final simplification 22.3%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))