Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.4% → 79.6%

Time: 35.0s

Precision: binary64

Cost: 40200

• Unsound rule application detected in e-graph. Results may not be sound.

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+215}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{D}{\sqrt{\ell}}\right)}^{2} \cdot 0.125\right)\\ \end{array} \]

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)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l))))
   (if (<= l -2e-310)
     (* (* (pow (* (pow (/ -1.0 h) 0.25) (pow (- d) 0.25)) 2.0) t_0) t_1)
     (if (<= l 5e+215)
       (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))
       (*
        (* (sqrt (/ d h)) (* (sqrt d) (sqrt (/ 1.0 l))))
        (-
         1.0
         (* (pow (* (/ (sqrt h) (/ d M)) (/ D (sqrt l))) 2.0) 0.125)))))))

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)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = 1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l);
	double tmp;
	if (l <= -2e-310) {
		tmp = (pow((pow((-1.0 / h), 0.25) * pow(-d, 0.25)), 2.0) * t_0) * t_1;
	} else if (l <= 5e+215) {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = (sqrt((d / h)) * (sqrt(d) * sqrt((1.0 / l)))) * (1.0 - (pow(((sqrt(h) / (d / M)) * (D / sqrt(l))), 2.0) * 0.125));
	}
	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
    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

↓

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 = 1.0d0 - ((h * (0.5d0 * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0))) / l)
    if (l <= (-2d-310)) then
        tmp = ((((((-1.0d0) / h) ** 0.25d0) * (-d ** 0.25d0)) ** 2.0d0) * t_0) * t_1
    else if (l <= 5d+215) then
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    else
        tmp = (sqrt((d / h)) * (sqrt(d) * sqrt((1.0d0 / l)))) * (1.0d0 - ((((sqrt(h) / (d / m)) * (d_1 / sqrt(l))) ** 2.0d0) * 0.125d0))
    end if
    code = tmp
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)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 1.0 - ((h * (0.5 * Math.pow((M * (D / (d * 2.0))), 2.0))) / l);
	double tmp;
	if (l <= -2e-310) {
		tmp = (Math.pow((Math.pow((-1.0 / h), 0.25) * Math.pow(-d, 0.25)), 2.0) * t_0) * t_1;
	} else if (l <= 5e+215) {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(d) * Math.sqrt((1.0 / l)))) * (1.0 - (Math.pow(((Math.sqrt(h) / (d / M)) * (D / Math.sqrt(l))), 2.0) * 0.125));
	}
	return tmp;
}

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)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = 1.0 - ((h * (0.5 * math.pow((M * (D / (d * 2.0))), 2.0))) / l)
	tmp = 0
	if l <= -2e-310:
		tmp = (math.pow((math.pow((-1.0 / h), 0.25) * math.pow(-d, 0.25)), 2.0) * t_0) * t_1
	elif l <= 5e+215:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = (math.sqrt((d / h)) * (math.sqrt(d) * math.sqrt((1.0 / l)))) * (1.0 - (math.pow(((math.sqrt(h) / (d / M)) * (D / math.sqrt(l))), 2.0) * 0.125))
	return tmp

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64((Float64((Float64(-1.0 / h) ^ 0.25) * (Float64(-d) ^ 0.25)) ^ 2.0) * t_0) * t_1);
	elseif (l <= 5e+215)
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) * sqrt(Float64(1.0 / l)))) * Float64(1.0 - Float64((Float64(Float64(sqrt(h) / Float64(d / M)) * Float64(D / sqrt(l))) ^ 2.0) * 0.125)));
	end
	return tmp
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

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = 1.0 - ((h * (0.5 * ((M * (D / (d * 2.0))) ^ 2.0))) / l);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (((((-1.0 / h) ^ 0.25) * (-d ^ 0.25)) ^ 2.0) * t_0) * t_1;
	elseif (l <= 5e+215)
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	else
		tmp = (sqrt((d / h)) * (sqrt(d) * sqrt((1.0 / l)))) * (1.0 - ((((sqrt(h) / (d / M)) * (D / sqrt(l))) ^ 2.0) * 0.125));
	end
	tmp_2 = tmp;
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]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Power[N[(N[Power[N[(-1.0 / h), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[(-d), 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 5e+215], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(N[Sqrt[h], $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 65.2%

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

   2. Simplified 63.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      Step-by-step derivation

      [Start] 65.2%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 65.2%	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 65.2%	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 63.6%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 63.6%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
      Step-by-step derivation

      [Start] 63.6%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      associate-*r* [=>] 63.6%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]
      frac-times [=>] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [<=] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]
      metadata-eval [<=] 65.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*r/ [=>] 67.1%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
      metadata-eval [=>] 67.1%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]
      *-commutative [=>] 67.1%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]
      frac-times [<=] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-*l/ [=>] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-*r/ [<=] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-/l/ [=>] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      *-commutative [=>] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   4. Applied egg-rr 65.4%

      \[\leadsto \left(\color{blue}{{\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      Step-by-step derivation

      [Start] 65.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      pow1/2 [=>] 65.5%	\[ \left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      sqr-pow [=>] 65.4%	\[ \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      pow2 [=>] 65.4%	\[ \left(\color{blue}{{\left({\left(\frac{d}{h}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      metadata-eval [=>] 65.4%	\[ \left({\left({\left(\frac{d}{h}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Taylor expanded in h around -inf 74.3%

      \[\leadsto \left({\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{h}\right) + \log \left(-1 \cdot d\right)\right)}\right)}}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Simplified 77.4%

      \[\leadsto \left({\color{blue}{\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      Step-by-step derivation

      [Start] 74.3%	\[ \left({\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{h}\right) + \log \left(-1 \cdot d\right)\right)}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      distribute-rgt-in [=>] 74.3%	\[ \left({\left(e^{\color{blue}{\log \left(\frac{-1}{h}\right) \cdot 0.25 + \log \left(-1 \cdot d\right) \cdot 0.25}}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      exp-sum [=>] 74.3%	\[ \left({\color{blue}{\left(e^{\log \left(\frac{-1}{h}\right) \cdot 0.25} \cdot e^{\log \left(-1 \cdot d\right) \cdot 0.25}\right)}}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      exp-to-pow [=>] 75.1%	\[ \left({\left(\color{blue}{{\left(\frac{-1}{h}\right)}^{0.25}} \cdot e^{\log \left(-1 \cdot d\right) \cdot 0.25}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      exp-to-pow [=>] 77.4%	\[ \left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot \color{blue}{{\left(-1 \cdot d\right)}^{0.25}}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      mul-1-neg [=>] 77.4%	\[ \left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\color{blue}{\left(-d\right)}}^{0.25}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -1.999999999999994e-310 < l < 5.0000000000000001e215

   1. Initial program 71.8%

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

   2. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      Step-by-step derivation

      [Start] 71.8%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 71.8%	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 71.8%	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 75.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
      Step-by-step derivation

      [Start] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      associate-*r* [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]
      frac-times [=>] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [<=] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]
      metadata-eval [<=] 71.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*r/ [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
      metadata-eval [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]
      *-commutative [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]
      frac-times [<=] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-*l/ [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-*r/ [<=] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      associate-/l/ [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      *-commutative [=>] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   4. Applied egg-rr 88.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      Step-by-step derivation

      [Start] 75.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
      sqrt-div [=>] 88.1%	\[ \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 5.0000000000000001e215 < l

   1. Initial program 54.3%

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

   2. Simplified 54.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      Step-by-step derivation

      [Start] 54.3%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 54.3%	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 54.3%	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 77.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      Step-by-step derivation

      [Start] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      pow1/2 [=>] 54.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      div-inv [=>] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot {\color{blue}{\left(d \cdot \frac{1}{\ell}\right)}}^{0.5}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      unpow-prod-down [=>] 77.0%	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({d}^{0.5} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      pow1/2 [<=] 77.0%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{d}} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   4. Simplified 77.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      Step-by-step derivation

      [Start] 77.0%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 77.0%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}\right)\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Taylor expanded in M around 0 54.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]

   6. Simplified 76.8%

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

      [Start] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [=>] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]
      *-commutative [=>] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [<=] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]
      *-commutative [=>] 54.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]
      times-frac [=>] 54.4%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]
      unpow2 [=>] 54.4%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]
      associate-/l* [=>] 63.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\frac{h}{\frac{{d}^{2}}{{M}^{2}}}}\right) \cdot 0.125\right) \]
      unpow2 [=>] 63.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{\color{blue}{d \cdot d}}{{M}^{2}}}\right) \cdot 0.125\right) \]
      unpow2 [=>] 63.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d \cdot d}{\color{blue}{M \cdot M}}}\right) \cdot 0.125\right) \]
      times-frac [=>] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\color{blue}{\frac{d}{M} \cdot \frac{d}{M}}}\right) \cdot 0.125\right) \]

   7. Applied egg-rr 90.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{{\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{D}{\sqrt{\ell}}\right)}^{2}} \cdot 0.125\right) \]
      Step-by-step derivation

      [Start] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right) \cdot 0.125\right) \]
      add-sqr-sqrt [=>] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\sqrt{\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}} \cdot \sqrt{\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}}\right)} \cdot 0.125\right) \]
      pow2 [=>] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}}\right)}^{2}} \cdot 0.125\right) \]
      *-commutative [=>] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D \cdot D}{\ell}}}\right)}^{2} \cdot 0.125\right) \]
      sqrt-prod [=>] 76.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{h}{\frac{d}{M} \cdot \frac{d}{M}}} \cdot \sqrt{\frac{D \cdot D}{\ell}}\right)}}^{2} \cdot 0.125\right) \]
      sqrt-div [=>] 77.2%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\color{blue}{\frac{\sqrt{h}}{\sqrt{\frac{d}{M} \cdot \frac{d}{M}}}} \cdot \sqrt{\frac{D \cdot D}{\ell}}\right)}^{2} \cdot 0.125\right) \]
      sqrt-prod [=>] 31.8%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\color{blue}{\sqrt{\frac{d}{M}} \cdot \sqrt{\frac{d}{M}}}} \cdot \sqrt{\frac{D \cdot D}{\ell}}\right)}^{2} \cdot 0.125\right) \]
      add-sqr-sqrt [<=] 86.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\color{blue}{\frac{d}{M}}} \cdot \sqrt{\frac{D \cdot D}{\ell}}\right)}^{2} \cdot 0.125\right) \]
      sqrt-div [=>] 86.3%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \color{blue}{\frac{\sqrt{D \cdot D}}{\sqrt{\ell}}}\right)}^{2} \cdot 0.125\right) \]
      sqrt-prod [=>] 54.5%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{\color{blue}{\sqrt{D} \cdot \sqrt{D}}}{\sqrt{\ell}}\right)}^{2} \cdot 0.125\right) \]
      add-sqr-sqrt [<=] 90.7%	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{\color{blue}{D}}{\sqrt{\ell}}\right)}^{2} \cdot 0.125\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{D}{\sqrt{\ell}}\right)}^{2} \cdot 0.125\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.6%
Cost	40200

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+215}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\frac{d}{M}} \cdot \frac{D}{\sqrt{\ell}}\right)}^{2} \cdot 0.125\right)\\ \end{array} \]

Alternative 2
Accuracy	79.2%
Cost	34052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+215}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \frac{M}{d}}{\frac{\frac{\ell \cdot \frac{d}{M}}{D}}{D}}\right)\\ \end{array} \]

Alternative 3
Accuracy	75.4%
Cost	27532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-118}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_1 \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-281}:\\ \;\;\;\;\left(t_0 \cdot t_2\right) \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{+215}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot t_1\right)}{\ell}\right) \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \frac{M}{d}}{\frac{\frac{\ell \cdot \frac{d}{M}}{D}}{D}}\right)\\ \end{array} \]

Alternative 4
Accuracy	74.8%
Cost	27532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -7.9 \cdot 10^{-118}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-281}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+215}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \frac{M}{d}}{\frac{\frac{\ell \cdot \frac{d}{M}}{D}}{D}}\right)\\ \end{array} \]

Alternative 5
Accuracy	71.2%
Cost	21320

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1.52 \cdot 10^{-118}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+210}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_0\right) \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \frac{M}{d}}{\frac{\frac{\ell \cdot \frac{d}{M}}{D}}{D}}\right)\\ \end{array} \]

Alternative 6
Accuracy	71.3%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-118}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 9.4 \cdot 10^{+213}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	71.4%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-119}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+213}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	70.7%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-117}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot D}{\frac{d}{h}}\right) \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	66.4%
Cost	14792

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-256}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+93}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\frac{0.125 \cdot \left(D \cdot D\right)}{d} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	55.9%
Cost	14600

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+158}:\\ \;\;\;\;\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	57.7%
Cost	14600

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	65.3%
Cost	14600

\[\begin{array}{l} t_0 := {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{-256}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0 \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - t_0 \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	58.3%
Cost	14468

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.1 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	49.3%
Cost	13512

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{if}\;D \leq -7 \cdot 10^{-92}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{M \cdot \left(h \cdot M\right)}{\frac{\ell}{D \cdot D}} \cdot \frac{0.125}{d \cdot d}\right)\\ \mathbf{elif}\;D \leq 2.8 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	47.1%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16
Accuracy	47.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17
Accuracy	44.2%
Cost	8260

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.65 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Accuracy	26.7%
Cost	6784

\[d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Alternative 19
Accuracy	26.7%
Cost	6720

\[\frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023271 
(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)))))