Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 18.4

Time: 46.6s

Precision: binary64

Cost: 27396

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-116}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-247}:\\ \;\;\;\;\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{M \cdot D}} - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-\ell}} \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

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 (sqrt (/ d h))) (t_2 (sqrt (- d))))
   (if (<= d -1.8e+102)
     (*
      (* (/ t_2 (sqrt (- h))) t_0)
      (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
     (if (<= d -5e-116)
       (*
        t_1
        (* t_0 (+ 1.0 (* -0.125 (/ (/ (* D (/ M d)) (/ (/ l D) (* h M))) d)))))
       (if (<= d -3.6e-247)
         (-
          (/ (* D (* M (sqrt (* (* h (pow l -3.0)) 0.015625)))) (/ d (* M D)))
          (* d (sqrt (/ (/ 1.0 l) h))))
         (if (<= d -2e-310)
           (*
            t_1
            (*
             (/ t_2 (sqrt (- l)))
             (+ 1.0 (* -0.125 (/ (* (* D (/ D l)) (* h (* M (/ M d)))) d)))))
           (*
            (+ 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 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= -1.8e+102) {
		tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	} else if (d <= -5e-116) {
		tmp = t_1 * (t_0 * (1.0 + (-0.125 * (((D * (M / d)) / ((l / D) / (h * M))) / d))));
	} else if (d <= -3.6e-247) {
		tmp = ((D * (M * sqrt(((h * pow(l, -3.0)) * 0.015625)))) / (d / (M * D))) - (d * sqrt(((1.0 / l) / h)));
	} else if (d <= -2e-310) {
		tmp = t_1 * ((t_2 / sqrt(-l)) * (1.0 + (-0.125 * (((D * (D / l)) * (h * (M * (M / d)))) / d))));
	} 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 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    if (d <= (-1.8d+102)) then
        tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))))
    else if (d <= (-5d-116)) then
        tmp = t_1 * (t_0 * (1.0d0 + ((-0.125d0) * (((d_1 * (m / d)) / ((l / d_1) / (h * m))) / d))))
    else if (d <= (-3.6d-247)) then
        tmp = ((d_1 * (m * sqrt(((h * (l ** (-3.0d0))) * 0.015625d0)))) / (d / (m * d_1))) - (d * sqrt(((1.0d0 / l) / h)))
    else if (d <= (-2d-310)) then
        tmp = t_1 * ((t_2 / sqrt(-l)) * (1.0d0 + ((-0.125d0) * (((d_1 * (d_1 / l)) * (h * (m * (m / d)))) / d))))
    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 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= -1.8e+102) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_0) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	} else if (d <= -5e-116) {
		tmp = t_1 * (t_0 * (1.0 + (-0.125 * (((D * (M / d)) / ((l / D) / (h * M))) / d))));
	} else if (d <= -3.6e-247) {
		tmp = ((D * (M * Math.sqrt(((h * Math.pow(l, -3.0)) * 0.015625)))) / (d / (M * D))) - (d * Math.sqrt(((1.0 / l) / h)));
	} else if (d <= -2e-310) {
		tmp = t_1 * ((t_2 / Math.sqrt(-l)) * (1.0 + (-0.125 * (((D * (D / l)) * (h * (M * (M / d)))) / d))));
	} 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 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= -1.8e+102:
		tmp = ((t_2 / math.sqrt(-h)) * t_0) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))
	elif d <= -5e-116:
		tmp = t_1 * (t_0 * (1.0 + (-0.125 * (((D * (M / d)) / ((l / D) / (h * M))) / d))))
	elif d <= -3.6e-247:
		tmp = ((D * (M * math.sqrt(((h * math.pow(l, -3.0)) * 0.015625)))) / (d / (M * D))) - (d * math.sqrt(((1.0 / l) / h)))
	elif d <= -2e-310:
		tmp = t_1 * ((t_2 / math.sqrt(-l)) * (1.0 + (-0.125 * (((D * (D / l)) * (h * (M * (M / d)))) / d))))
	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 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.8e+102)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))));
	elseif (d <= -5e-116)
		tmp = Float64(t_1 * Float64(t_0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * Float64(M / d)) / Float64(Float64(l / D) / Float64(h * M))) / d)))));
	elseif (d <= -3.6e-247)
		tmp = Float64(Float64(Float64(D * Float64(M * sqrt(Float64(Float64(h * (l ^ -3.0)) * 0.015625)))) / Float64(d / Float64(M * D))) - Float64(d * sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (d <= -2e-310)
		tmp = Float64(t_1 * Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * Float64(D / l)) * Float64(h * Float64(M * Float64(M / d)))) / d)))));
	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 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= -1.8e+102)
		tmp = ((t_2 / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))));
	elseif (d <= -5e-116)
		tmp = t_1 * (t_0 * (1.0 + (-0.125 * (((D * (M / d)) / ((l / D) / (h * M))) / d))));
	elseif (d <= -3.6e-247)
		tmp = ((D * (M * sqrt(((h * (l ^ -3.0)) * 0.015625)))) / (d / (M * D))) - (d * sqrt(((1.0 / l) / h)));
	elseif (d <= -2e-310)
		tmp = t_1 * ((t_2 / sqrt(-l)) * (1.0 + (-0.125 * (((D * (D / l)) * (h * (M * (M / d)))) / d))));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.8e+102], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-116], N[(t$95$1 * N[(t$95$0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.6e-247], N[(N[(N[(D * N[(M * N[Sqrt[N[(N[(h * N[Power[l, -3.0], $MachinePrecision]), $MachinePrecision] * 0.015625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(t$95$1 * N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $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 d < -1.8000000000000001e102

   1. Initial program 26.6

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

   2. Simplified 25.9

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

      [Start] 26.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 26.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 26.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 25.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 12.1

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

   if -1.8000000000000001e102 < d < -5.0000000000000003e-116

   1. Initial program 18.9

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

   2. Simplified 19.8

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

      [Start] 18.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 19.2	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 19.2	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 19.2	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 19.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Taylor expanded in M around 0 31.0

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

   4. Simplified 30.3

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

      [Start] 31.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 31.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 31.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]
      times-frac [=>] 30.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 30.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]
      *-commutative [=>] 30.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 30.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 30.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)\right) \]

   5. Applied egg-rr 27.1

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

   6. Applied egg-rr 17.2

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

   if -5.0000000000000003e-116 < d < -3.5999999999999997e-247

   1. Initial program 35.2

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

   2. Simplified 36.3

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

      [Start] 35.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 35.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 35.2	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 35.2	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 35.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 35.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 35.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 36.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 36.3

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

   4. Taylor expanded in d around -inf 37.7

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

   5. Simplified 36.3

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

      [Start] 37.7	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      mul-1-neg [=>] 37.7	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
      unsub-neg [=>] 37.7	\[ \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 37.7	\[ \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-*l* [=>] 37.7	\[ \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l* [=>] 39.4	\[ \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 39.4	\[ \frac{{D}^{2}}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 38.6	\[ \frac{{D}^{2}}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r/ [=>] 36.4	\[ \color{blue}{\left(\frac{{D}^{2}}{\frac{d}{M}} \cdot M\right)} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 36.4	\[ \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{M}} \cdot M\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 36.3	\[ \left(\frac{D \cdot D}{\frac{d}{M}} \cdot M\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Applied egg-rr 31.2

      \[\leadsto \color{blue}{\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{D \cdot M}}} - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

   if -3.5999999999999997e-247 < d < -1.999999999999994e-310

   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 45.0

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

      [Start] 44.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 44.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 44.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 44.0	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 44.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Taylor expanded in M around 0 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]
      times-frac [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]
      *-commutative [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)\right) \]

   5. Applied egg-rr 45.9

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

   6. Applied egg-rr 38.3

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

   if -1.999999999999994e-310 < d

   1. Initial program 26.9

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

   2. Simplified 27.3

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

      [Start] 26.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 27.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 16.6

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

   4. Simplified 16.6

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

      [Start] 16.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-lft-identity [<=] 16.6	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [<=] 16.6	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 16.6	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      *-commutative [=>] 16.6	\[ \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [=>] 16.6	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 16.6	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-247}:\\ \;\;\;\;\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{M \cdot D}} - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.2
Cost	21584

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+87}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-117}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{M \cdot D}} - d \cdot t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2
Error	20.4
Cost	21188

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-39}:\\ \;\;\;\;\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{M \cdot D}} - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + 0.125 \cdot \left(\frac{D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}} \cdot \frac{D}{-\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Error	21.9
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq -1.22 \cdot 10^{+126}:\\ \;\;\;\;\left(t_0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-39}:\\ \;\;\;\;\left(M \cdot \left(D \cdot \frac{M}{\frac{d}{D}}\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot t_1\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + 0.125 \cdot \left(\frac{D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}} \cdot \frac{D}{-\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Error	21.3
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+231}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+127}:\\ \;\;\;\;\left(t_0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-39}:\\ \;\;\;\;\frac{D \cdot \left(M \cdot \sqrt{\left(h \cdot {\ell}^{-3}\right) \cdot 0.015625}\right)}{\frac{d}{M \cdot D}} - d \cdot t_1\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + 0.125 \cdot \left(\frac{D}{\frac{d}{h} \cdot \frac{d}{M \cdot M}} \cdot \frac{D}{-\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	21.9
Cost	21004

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+232}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot t_0\\ \end{array} \]

Alternative 6
Error	23.0
Cost	14996

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+86}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-269}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	23.0
Cost	14996

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+86}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -8.4 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-269}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	22.8
Cost	14996

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+88}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(M \cdot M\right)}{\frac{\ell}{D} \cdot \left(d \cdot \frac{d}{h}\right)}\right)\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-269}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Error	22.4
Cost	14924

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-91}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\frac{\ell}{D} \cdot \left(d \cdot d\right)}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Error	22.1
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+87}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\frac{\ell}{D} \cdot \left(d \cdot d\right)}\right)\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-273}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	21.0
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+87}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Error	22.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Error	23.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Error	27.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 15
Error	27.2
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	34.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Error	33.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	33.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 1.75 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 19
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2023060 
(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)))))