Henrywood and Agarwal, Equation (12)

Average Error: 26.2 → 15.4

Time: 1.0min

Precision: binary64

Cost: 40588

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq 0:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 10^{-96}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\sqrt{h} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

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 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
        (t_1 (pow (/ d h) 0.5))
        (t_2 (sqrt (- d))))
   (if (<= d 0.0)
     (* (* (/ t_2 (sqrt (- h))) (/ t_2 (sqrt (- l)))) t_0)
     (if (<= d 1e-96)
       (* t_0 (* t_1 (/ (sqrt d) (sqrt l))))
       (if (<= d 5e+53)
         (*
          (* t_1 (pow (/ d l) 0.5))
          (-
           1.0
           (pow
            (*
             (* (sqrt h) (/ 1.0 (sqrt l)))
             (* (* (/ M 2.0) (/ D d)) (sqrt 0.5)))
            2.0)))
         (/ 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 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l));
	double t_1 = pow((d / h), 0.5);
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= 0.0) {
		tmp = ((t_2 / sqrt(-h)) * (t_2 / sqrt(-l))) * t_0;
	} else if (d <= 1e-96) {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	} else if (d <= 5e+53) {
		tmp = (t_1 * pow((d / l), 0.5)) * (1.0 - pow(((sqrt(h) * (1.0 / sqrt(l))) * (((M / 2.0) * (D / d)) * sqrt(0.5))), 2.0));
	} else {
		tmp = 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 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l))
    t_1 = (d / h) ** 0.5d0
    t_2 = sqrt(-d)
    if (d <= 0.0d0) then
        tmp = ((t_2 / sqrt(-h)) * (t_2 / sqrt(-l))) * t_0
    else if (d <= 1d-96) then
        tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)))
    else if (d <= 5d+53) then
        tmp = (t_1 * ((d / l) ** 0.5d0)) * (1.0d0 - (((sqrt(h) * (1.0d0 / sqrt(l))) * (((m / 2.0d0) * (d_1 / d)) * sqrt(0.5d0))) ** 2.0d0))
    else
        tmp = 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 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l));
	double t_1 = Math.pow((d / h), 0.5);
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= 0.0) {
		tmp = ((t_2 / Math.sqrt(-h)) * (t_2 / Math.sqrt(-l))) * t_0;
	} else if (d <= 1e-96) {
		tmp = t_0 * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	} else if (d <= 5e+53) {
		tmp = (t_1 * Math.pow((d / l), 0.5)) * (1.0 - Math.pow(((Math.sqrt(h) * (1.0 / Math.sqrt(l))) * (((M / 2.0) * (D / d)) * Math.sqrt(0.5))), 2.0));
	} else {
		tmp = 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 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l))
	t_1 = math.pow((d / h), 0.5)
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= 0.0:
		tmp = ((t_2 / math.sqrt(-h)) * (t_2 / math.sqrt(-l))) * t_0
	elif d <= 1e-96:
		tmp = t_0 * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	elif d <= 5e+53:
		tmp = (t_1 * math.pow((d / l), 0.5)) * (1.0 - math.pow(((math.sqrt(h) * (1.0 / math.sqrt(l))) * (((M / 2.0) * (D / d)) * math.sqrt(0.5))), 2.0))
	else:
		tmp = 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(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(d / h) ^ 0.5
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= 0.0)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(t_2 / sqrt(Float64(-l)))) * t_0);
	elseif (d <= 1e-96)
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	elseif (d <= 5e+53)
		tmp = Float64(Float64(t_1 * (Float64(d / l) ^ 0.5)) * Float64(1.0 - (Float64(Float64(sqrt(h) * Float64(1.0 / sqrt(l))) * Float64(Float64(Float64(M / 2.0) * Float64(D / d)) * sqrt(0.5))) ^ 2.0)));
	else
		tmp = 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 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l));
	t_1 = (d / h) ^ 0.5;
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= 0.0)
		tmp = ((t_2 / sqrt(-h)) * (t_2 / sqrt(-l))) * t_0;
	elseif (d <= 1e-96)
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	elseif (d <= 5e+53)
		tmp = (t_1 * ((d / l) ^ 0.5)) * (1.0 - (((sqrt(h) * (1.0 / sqrt(l))) * (((M / 2.0) * (D / d)) * sqrt(0.5))) ^ 2.0));
	else
		tmp = 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[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, 0.0], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 1e-96], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+53], N[(N[(t$95$1 * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(N[Sqrt[h], $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < 0.0

   1. Initial program 26.2

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

   2. Applied egg-rr 22.1

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

   3. Applied egg-rr 16.3

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

   if 0.0 < d < 9.9999999999999991e-97

   1. Initial program 36.0

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

   2. Applied egg-rr 30.1

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

   if 9.9999999999999991e-97 < d < 5.0000000000000004e53

   1. Initial program 16.7

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

   2. Applied egg-rr 14.7

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

   3. Applied egg-rr 9.7

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

   if 5.0000000000000004e53 < d

   1. Initial program 25.5

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

   2. Taylor expanded in d around inf 18.4

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

   3. Applied egg-rr 18.3

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

   4. Applied egg-rr 7.1

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

4. Final simplification 15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 0:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 10^{-96}:\\ \;\;\;\;\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\sqrt{h} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.3
Cost	104464

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := t_1 \cdot \left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ t_4 := t_1 \cdot \left(t_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-223}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;t_3 \leq 10^{+292}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 2
Error	15.4
Cost	40460

\[\begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq 0:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 10^{-96}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Error	16.7
Cost	33924

\[\begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq 0:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 10^{-96}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 10^{+60}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{\ell} \cdot \frac{D}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 10^{+105}:\\ \;\;\;\;t_0 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Error	18.4
Cost	33668

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_3}{\sqrt{-\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_1 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right)\\ \mathbf{elif}\;h \leq 10^{-52}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{+197}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	19.4
Cost	27992

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t_2}{\sqrt{-\ell}}\\ t_4 := \frac{D}{\frac{d}{D}}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot t_5\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{{\ell}^{3}}}, 0.125 \cdot \left(M \cdot \left(M \cdot t_4\right)\right), \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\left(t_3 \cdot t_0\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{t_4}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;\left(t_0 \cdot t_5\right) \cdot \left(1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{-145}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

Alternative 6
Error	18.4
Cost	27860

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot t_1\right)\\ \mathbf{elif}\;h \leq 10^{-52}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{+197}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	19.8
Cost	27528

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot t_1\right)\\ \mathbf{elif}\;h \leq 10^{-52}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+117}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + \left(M \cdot \frac{\frac{M \cdot \left(D \cdot \frac{D}{d}\right)}{\ell}}{\frac{d}{h}}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 8
Error	20.4
Cost	27396

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{t_2}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;h \leq 10^{-52}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+117}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + \left(M \cdot \frac{\frac{M \cdot \left(D \cdot \frac{D}{d}\right)}{\ell}}{\frac{d}{h}}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 9
Error	22.2
Cost	21000

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_0\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{\ell} \cdot \frac{D}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Error	22.0
Cost	15780

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := t_1 \cdot \left(1 + \left(M \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{\ell} \cdot \frac{D}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.25 \cdot 10^{-142}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \left(M \cdot \frac{\left(D \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{d \cdot \ell}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-257}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-238}:\\ \;\;\;\;\left(D \cdot \left(M \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 10^{-80}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 10^{-23}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.7
Cost	15384

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_2\right) \cdot \left(1 + -0.125 \cdot \left(M \cdot \frac{D \cdot \frac{D}{d}}{\frac{\frac{d \cdot \ell}{h}}{M}}\right)\right)\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -9.8 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-299}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right) \cdot \left(t_2 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.8
Cost	15252

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	22.0
Cost	15252

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-103}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(M \cdot \frac{M \cdot D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(M \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{\ell} \cdot \frac{D}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	22.1
Cost	15252

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-151}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(M \cdot \frac{\frac{M \cdot \left(D \cdot \frac{D}{d}\right)}{\ell}}{\frac{d}{h}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+53}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(M \cdot \left(M \cdot \left(\frac{\frac{D}{d}}{\ell} \cdot \frac{D}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	24.2
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;h \leq -1.2 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 10^{-50}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+200}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	24.3
Cost	15056

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.2 \cdot 10^{+189}:\\ \;\;\;\;t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 10^{-50}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+200}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right) \cdot \left(t_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Error	22.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;h \leq -1.2 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 18
Error	22.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.2 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 19
Error	22.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 20
Error	27.2
Cost	7044

\[\begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 21
Error	36.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	36.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 10^{-300}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 23
Error	43.7
Cost	6784

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

Alternative 24
Error	43.7
Cost	6720

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

Reproduce

herbie shell --seed 2022300 
(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)))))