Henrywood and Agarwal, Equation (12)

Average Accuracy: 59.0% → 75.0%

Time: 54.2s

Precision: binary64

Cost: 27856

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \frac{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\\ t_1 := \sqrt{-d}\\ t_2 := \left(\frac{t_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_0\\ \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -14000000000:\\ \;\;\;\;\frac{0.125}{d} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{t_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

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 (- 1.0 (/ (* 0.5 (* h (pow (* 0.5 (/ (* M D) d)) 2.0))) l)))
        (t_1 (sqrt (- d)))
        (t_2 (* (* (/ t_1 (sqrt (- h))) (pow (/ d l) 0.5)) t_0)))
   (if (<= l -1.22e+58)
     t_2
     (if (<= l -14000000000.0)
       (-
        (* (/ 0.125 d) (* (pow (* M D) 2.0) (sqrt (* h (pow l -3.0)))))
        (* d (sqrt (/ 1.0 (* l h)))))
       (if (<= l -8.6e-115)
         t_2
         (if (<= l -2e-310)
           (* t_0 (* (pow (/ d h) 0.5) (/ t_1 (sqrt (- l)))))
           (if (<= l 8.2e-156)
             (* d (/ (/ -1.0 (sqrt l)) (- (sqrt h))))
             (*
              (+ 1.0 (* (pow (* M (* 0.5 (/ D d))) 2.0) (* (/ h l) -0.5)))
              (/ d (* (sqrt l) (sqrt h)))))))))))

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 = 1.0 - ((0.5 * (h * pow((0.5 * ((M * D) / d)), 2.0))) / l);
	double t_1 = sqrt(-d);
	double t_2 = ((t_1 / sqrt(-h)) * pow((d / l), 0.5)) * t_0;
	double tmp;
	if (l <= -1.22e+58) {
		tmp = t_2;
	} else if (l <= -14000000000.0) {
		tmp = ((0.125 / d) * (pow((M * D), 2.0) * sqrt((h * pow(l, -3.0))))) - (d * sqrt((1.0 / (l * h))));
	} else if (l <= -8.6e-115) {
		tmp = t_2;
	} else if (l <= -2e-310) {
		tmp = t_0 * (pow((d / h), 0.5) * (t_1 / sqrt(-l)));
	} else if (l <= 8.2e-156) {
		tmp = d * ((-1.0 / sqrt(l)) / -sqrt(h));
	} else {
		tmp = (1.0 + (pow((M * (0.5 * (D / d))), 2.0) * ((h / l) * -0.5))) * (d / (sqrt(l) * sqrt(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
    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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (h * ((0.5d0 * ((m * d_1) / d)) ** 2.0d0))) / l)
    t_1 = sqrt(-d)
    t_2 = ((t_1 / sqrt(-h)) * ((d / l) ** 0.5d0)) * t_0
    if (l <= (-1.22d+58)) then
        tmp = t_2
    else if (l <= (-14000000000.0d0)) then
        tmp = ((0.125d0 / d) * (((m * d_1) ** 2.0d0) * sqrt((h * (l ** (-3.0d0)))))) - (d * sqrt((1.0d0 / (l * h))))
    else if (l <= (-8.6d-115)) then
        tmp = t_2
    else if (l <= (-2d-310)) then
        tmp = t_0 * (((d / h) ** 0.5d0) * (t_1 / sqrt(-l)))
    else if (l <= 8.2d-156) then
        tmp = d * (((-1.0d0) / sqrt(l)) / -sqrt(h))
    else
        tmp = (1.0d0 + (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * ((h / l) * (-0.5d0)))) * (d / (sqrt(l) * sqrt(h)))
    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 = 1.0 - ((0.5 * (h * Math.pow((0.5 * ((M * D) / d)), 2.0))) / l);
	double t_1 = Math.sqrt(-d);
	double t_2 = ((t_1 / Math.sqrt(-h)) * Math.pow((d / l), 0.5)) * t_0;
	double tmp;
	if (l <= -1.22e+58) {
		tmp = t_2;
	} else if (l <= -14000000000.0) {
		tmp = ((0.125 / d) * (Math.pow((M * D), 2.0) * Math.sqrt((h * Math.pow(l, -3.0))))) - (d * Math.sqrt((1.0 / (l * h))));
	} else if (l <= -8.6e-115) {
		tmp = t_2;
	} else if (l <= -2e-310) {
		tmp = t_0 * (Math.pow((d / h), 0.5) * (t_1 / Math.sqrt(-l)));
	} else if (l <= 8.2e-156) {
		tmp = d * ((-1.0 / Math.sqrt(l)) / -Math.sqrt(h));
	} else {
		tmp = (1.0 + (Math.pow((M * (0.5 * (D / d))), 2.0) * ((h / l) * -0.5))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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 = 1.0 - ((0.5 * (h * math.pow((0.5 * ((M * D) / d)), 2.0))) / l)
	t_1 = math.sqrt(-d)
	t_2 = ((t_1 / math.sqrt(-h)) * math.pow((d / l), 0.5)) * t_0
	tmp = 0
	if l <= -1.22e+58:
		tmp = t_2
	elif l <= -14000000000.0:
		tmp = ((0.125 / d) * (math.pow((M * D), 2.0) * math.sqrt((h * math.pow(l, -3.0))))) - (d * math.sqrt((1.0 / (l * h))))
	elif l <= -8.6e-115:
		tmp = t_2
	elif l <= -2e-310:
		tmp = t_0 * (math.pow((d / h), 0.5) * (t_1 / math.sqrt(-l)))
	elif l <= 8.2e-156:
		tmp = d * ((-1.0 / math.sqrt(l)) / -math.sqrt(h))
	else:
		tmp = (1.0 + (math.pow((M * (0.5 * (D / d))), 2.0) * ((h / l) * -0.5))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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 = Float64(1.0 - Float64(Float64(0.5 * Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0))) / l))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5)) * t_0)
	tmp = 0.0
	if (l <= -1.22e+58)
		tmp = t_2;
	elseif (l <= -14000000000.0)
		tmp = Float64(Float64(Float64(0.125 / d) * Float64((Float64(M * D) ^ 2.0) * sqrt(Float64(h * (l ^ -3.0))))) - Float64(d * sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (l <= -8.6e-115)
		tmp = t_2;
	elseif (l <= -2e-310)
		tmp = Float64(t_0 * Float64((Float64(d / h) ^ 0.5) * Float64(t_1 / sqrt(Float64(-l)))));
	elseif (l <= 8.2e-156)
		tmp = Float64(d * Float64(Float64(-1.0 / sqrt(l)) / Float64(-sqrt(h))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(Float64(h / l) * -0.5))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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 = 1.0 - ((0.5 * (h * ((0.5 * ((M * D) / d)) ^ 2.0))) / l);
	t_1 = sqrt(-d);
	t_2 = ((t_1 / sqrt(-h)) * ((d / l) ^ 0.5)) * t_0;
	tmp = 0.0;
	if (l <= -1.22e+58)
		tmp = t_2;
	elseif (l <= -14000000000.0)
		tmp = ((0.125 / d) * (((M * D) ^ 2.0) * sqrt((h * (l ^ -3.0))))) - (d * sqrt((1.0 / (l * h))));
	elseif (l <= -8.6e-115)
		tmp = t_2;
	elseif (l <= -2e-310)
		tmp = t_0 * (((d / h) ^ 0.5) * (t_1 / sqrt(-l)));
	elseif (l <= 8.2e-156)
		tmp = d * ((-1.0 / sqrt(l)) / -sqrt(h));
	else
		tmp = (1.0 + (((M * (0.5 * (D / d))) ^ 2.0) * ((h / l) * -0.5))) * (d / (sqrt(l) * sqrt(h)));
	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[(1.0 - N[(N[(0.5 * N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[l, -1.22e+58], t$95$2, If[LessEqual[l, -14000000000.0], N[(N[(N[(0.125 / d), $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8.6e-115], t$95$2, If[LessEqual[l, -2e-310], N[(t$95$0 * N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.2e-156], N[(d * N[(N[(-1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[h], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.21999999999999995e58 or -1.4e10 < l < -8.6000000000000008e-115

   1. Initial program 61.9%

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

   2. Applied egg-rr 60.8%

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

      [Start] 61.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*r/ [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
      associate-*l* [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}}{\ell}\right) \]
      metadata-eval [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      *-un-lft-identity [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      times-frac [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]

   3. Applied egg-rr 71.9%

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

      [Start] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 60.8	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      unpow1/2 [=>] 60.8	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      frac-2neg [=>] 60.8	\[ \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 71.9	\[ \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]

   if -1.21999999999999995e58 < l < -1.4e10

   1. Initial program 72.0%

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

   2. Applied egg-rr 76.5%

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

      [Start] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      add-sqr-sqrt [=>] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]
      pow2 [=>] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. Applied egg-rr 76.1%

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

      [Start] 76.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      metadata-eval [=>] 76.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      unpow1/2 [=>] 76.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      clear-num [=>] 76.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      sqrt-div [=>] 76.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      metadata-eval [=>] 76.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   4. Taylor expanded in d around -inf 57.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) + 0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}} - d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
      Proof

      [Start] 57.9	\[ -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) + 0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      +-commutative [=>] 57.9	\[ \color{blue}{0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
      *-commutative [=>] 57.9	\[ 0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
      mul-1-neg [=>] 57.9	\[ 0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
      unsub-neg [=>] 57.9	\[ \color{blue}{0.25 \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - \sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{0.125}{d}} \cdot \left(D \cdot M\right)\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)}^{2}} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      Proof

      [Start] 64.2	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      add-sqr-sqrt [=>] 42.7	\[ \color{blue}{\sqrt{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}} \cdot \sqrt{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}}} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      pow2 [=>] 42.7	\[ \color{blue}{{\left(\sqrt{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}}\right)}^{2}} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{0.125}{d} \cdot \left({\left(D \cdot M\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      Proof

      [Start] 0.0	\[ {\left(\left(\sqrt{\frac{0.125}{d}} \cdot \left(D \cdot M\right)\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)}^{2} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      unpow2 [=>] 0.0	\[ \color{blue}{\left(\left(\sqrt{\frac{0.125}{d}} \cdot \left(D \cdot M\right)\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right) \cdot \left(\left(\sqrt{\frac{0.125}{d}} \cdot \left(D \cdot M\right)\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      associate-*l* [=>] 0.0	\[ \color{blue}{\left(\sqrt{\frac{0.125}{d}} \cdot \left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right)} \cdot \left(\left(\sqrt{\frac{0.125}{d}} \cdot \left(D \cdot M\right)\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      associate-*l* [=>] 0.0	\[ \left(\sqrt{\frac{0.125}{d}} \cdot \left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{0.125}{d}} \cdot \left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      swap-sqr [=>] 0.0	\[ \color{blue}{\left(\sqrt{\frac{0.125}{d}} \cdot \sqrt{\frac{0.125}{d}}\right) \cdot \left(\left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right) \cdot \left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      rem-square-sqrt [=>] 84.2	\[ \color{blue}{\frac{0.125}{d}} \cdot \left(\left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right) \cdot \left(\left(D \cdot M\right) \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      swap-sqr [=>] 80.7	\[ \frac{0.125}{d} \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left({\left(h \cdot {\ell}^{-3}\right)}^{0.25} \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right)} - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      unpow2 [<=] 80.7	\[ \frac{0.125}{d} \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left({\left(h \cdot {\ell}^{-3}\right)}^{0.25} \cdot {\left(h \cdot {\ell}^{-3}\right)}^{0.25}\right)\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      pow-sqr [=>] 80.7	\[ \frac{0.125}{d} \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{{\left(h \cdot {\ell}^{-3}\right)}^{\left(2 \cdot 0.25\right)}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      metadata-eval [=>] 80.7	\[ \frac{0.125}{d} \cdot \left({\left(D \cdot M\right)}^{2} \cdot {\left(h \cdot {\ell}^{-3}\right)}^{\color{blue}{0.5}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
      unpow1/2 [=>] 80.7	\[ \frac{0.125}{d} \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\sqrt{h \cdot {\ell}^{-3}}}\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   if -8.6000000000000008e-115 < l < -1.999999999999994e-310

   1. Initial program 49.9%

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

   2. Applied egg-rr 59.2%

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

      [Start] 49.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*r/ [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
      associate-*l* [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}}{\ell}\right) \]
      metadata-eval [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      *-un-lft-identity [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      times-frac [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]

   3. Applied egg-rr 83.6%

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

      [Start] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      unpow1/2 [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      frac-2neg [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 83.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h\right)}{\ell}\right) \]

   if -1.999999999999994e-310 < l < 8.2000000000000004e-156

   1. Initial program 44.0%

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

   2. Simplified 43.7%

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

      [Start] 44.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 44.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 44.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 44.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 44.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 44.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 44.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 43.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 42.3%

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

   4. Simplified 42.2%

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

      [Start] 42.3	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 42.3	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 42.2	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 65.7%

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

      [Start] 42.2	\[ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
      sqrt-div [=>] 65.7	\[ d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
      frac-2neg [=>] 65.7	\[ d \cdot \color{blue}{\frac{-\sqrt{\frac{1}{\ell}}}{-\sqrt{h}}} \]
      sqrt-div [=>] 65.7	\[ d \cdot \frac{-\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}}{-\sqrt{h}} \]
      metadata-eval [=>] 65.7	\[ d \cdot \frac{-\frac{\color{blue}{1}}{\sqrt{\ell}}}{-\sqrt{h}} \]
      distribute-neg-frac [=>] 65.7	\[ d \cdot \frac{\color{blue}{\frac{-1}{\sqrt{\ell}}}}{-\sqrt{h}} \]
      metadata-eval [=>] 65.7	\[ d \cdot \frac{\frac{\color{blue}{-1}}{\sqrt{\ell}}}{-\sqrt{h}} \]

   if 8.2000000000000004e-156 < l

   1. Initial program 59.6%

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

   2. Simplified 59.3%

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

      [Start] 59.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 59.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 76.6%

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

      [Start] 59.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 59.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      distribute-lft-in [=>] 59.3	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [<=] 59.3	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      *-un-lft-identity [<=] 59.3	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 59.9	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 64.1	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 64.1	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 64.3	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 74.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 76.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 76.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 76.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   4. Simplified 76.6%

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

      [Start] 76.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-lft-identity [<=] 76.6	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [<=] 76.6	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 76.6	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      *-commutative [=>] 76.6	\[ \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [=>] 76.6	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 76.6	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -14000000000:\\ \;\;\;\;\frac{0.125}{d} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -8.6 \cdot 10^{-115}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.1%
Cost	27724

\[\begin{array}{l} t_0 := \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{0.5 \cdot \left(h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -510000000000:\\ \;\;\;\;\frac{0.125}{d} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-155}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2
Accuracy	72.5%
Cost	27396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+57}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -40000000000:\\ \;\;\;\;\frac{0.125}{d} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_1}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-154}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Accuracy	69.5%
Cost	27208

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+180}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -850000000000:\\ \;\;\;\;\frac{0.125}{d} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \sqrt{h \cdot {\ell}^{-3}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t_2 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \frac{M \cdot \left(h \cdot D\right)}{\ell \cdot \frac{d}{D}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_1}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Accuracy	70.7%
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ t_3 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+233}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \frac{M \cdot \left(h \cdot D\right)}{\ell \cdot \frac{d}{D}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-235}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_3}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-271}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-154}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_3 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Accuracy	67.3%
Cost	21004

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+120}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{M}{\ell \cdot \frac{d \cdot d}{M \cdot \left(h \cdot D\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Accuracy	64.6%
Cost	21004

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+120}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+23}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \frac{M}{\left(d \cdot \frac{\frac{\ell}{D}}{D}\right) \cdot \frac{d}{h \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Accuracy	68.9%
Cost	21004

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+140}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-154}:\\ \;\;\;\;d \cdot \frac{\frac{-1}{\sqrt{\ell}}}{-\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Accuracy	67.2%
Cost	15056

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{M}{\ell \cdot \frac{d \cdot d}{M \cdot \left(h \cdot D\right)}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	64.6%
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+102}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1.36 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-277}:\\ \;\;\;\;\left(1 + \frac{-0.125}{\frac{\frac{\ell}{h}}{{\left(\frac{M \cdot D}{d}\right)}^{2}}}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Accuracy	67.5%
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M \cdot \left(h \cdot \frac{D}{\ell}\right)}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	62.7%
Cost	14472

\[\begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+115}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-294}:\\ \;\;\;\;\left(1 + \frac{-0.125}{\frac{\frac{\ell}{h}}{{\left(\frac{M \cdot D}{d}\right)}^{2}}}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Accuracy	63.2%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+100}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Accuracy	63.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.42 \cdot 10^{-284}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Accuracy	46.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-276} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+172}\right):\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 15
Accuracy	55.2%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-283}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]

Alternative 16
Accuracy	45.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 17
Accuracy	30.8%
Cost	6720

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

Reproduce

herbie shell --seed 2023137 
(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)))))