Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 17.1

Time: 13.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{\frac{\ell}{d}}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\\ t_3 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot t_2\right)}^{2}\\ t_4 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\left(\frac{t_4}{\sqrt{-h}} \cdot t_0\right) \cdot t_3\\ \mathbf{elif}\;d \leq 0:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot \frac{1}{\frac{\sqrt{-\ell}}{t_4}}\right)\\ \mathbf{elif}\;d \leq 10^{-50}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+200}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+303}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(t_2 \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (/ l d))))
        (t_1 (pow (/ d h) 0.5))
        (t_2 (* (/ M (/ (* d 2.0) D)) (sqrt 0.5)))
        (t_3 (- 1.0 (pow (* (sqrt (/ h l)) t_2) 2.0)))
        (t_4 (sqrt (- d))))
   (if (<= d -1e-105)
     (* (* (/ t_4 (sqrt (- h))) t_0) t_3)
     (if (<= d 0.0)
       (* t_3 (* t_1 (/ 1.0 (/ (sqrt (- l)) t_4))))
       (if (<= d 1e-50)
         (* t_3 (* t_1 (* (sqrt d) (sqrt (/ 1.0 l)))))
         (if (<= d 2.6e+200)
           (* t_3 (* t_0 (/ (sqrt d) (sqrt h))))
           (if (<= d 3.5e+303)
             (* d (sqrt (/ 1.0 (* h l))))
             (*
              (* t_1 (pow (/ d l) 0.5))
              (- 1.0 (pow (* t_2 (/ (sqrt h) (sqrt l))) 2.0))))))))))

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 / sqrt((l / d));
	double t_1 = pow((d / h), 0.5);
	double t_2 = (M / ((d * 2.0) / D)) * sqrt(0.5);
	double t_3 = 1.0 - pow((sqrt((h / l)) * t_2), 2.0);
	double t_4 = sqrt(-d);
	double tmp;
	if (d <= -1e-105) {
		tmp = ((t_4 / sqrt(-h)) * t_0) * t_3;
	} else if (d <= 0.0) {
		tmp = t_3 * (t_1 * (1.0 / (sqrt(-l) / t_4)));
	} else if (d <= 1e-50) {
		tmp = t_3 * (t_1 * (sqrt(d) * sqrt((1.0 / l))));
	} else if (d <= 2.6e+200) {
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	} else if (d <= 3.5e+303) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else {
		tmp = (t_1 * pow((d / l), 0.5)) * (1.0 - pow((t_2 * (sqrt(h) / sqrt(l))), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((l / d))
    t_1 = (d / h) ** 0.5d0
    t_2 = (m / ((d * 2.0d0) / d_1)) * sqrt(0.5d0)
    t_3 = 1.0d0 - ((sqrt((h / l)) * t_2) ** 2.0d0)
    t_4 = sqrt(-d)
    if (d <= (-1d-105)) then
        tmp = ((t_4 / sqrt(-h)) * t_0) * t_3
    else if (d <= 0.0d0) then
        tmp = t_3 * (t_1 * (1.0d0 / (sqrt(-l) / t_4)))
    else if (d <= 1d-50) then
        tmp = t_3 * (t_1 * (sqrt(d) * sqrt((1.0d0 / l))))
    else if (d <= 2.6d+200) then
        tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)))
    else if (d <= 3.5d+303) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    else
        tmp = (t_1 * ((d / l) ** 0.5d0)) * (1.0d0 - ((t_2 * (sqrt(h) / sqrt(l))) ** 2.0d0))
    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 / Math.sqrt((l / d));
	double t_1 = Math.pow((d / h), 0.5);
	double t_2 = (M / ((d * 2.0) / D)) * Math.sqrt(0.5);
	double t_3 = 1.0 - Math.pow((Math.sqrt((h / l)) * t_2), 2.0);
	double t_4 = Math.sqrt(-d);
	double tmp;
	if (d <= -1e-105) {
		tmp = ((t_4 / Math.sqrt(-h)) * t_0) * t_3;
	} else if (d <= 0.0) {
		tmp = t_3 * (t_1 * (1.0 / (Math.sqrt(-l) / t_4)));
	} else if (d <= 1e-50) {
		tmp = t_3 * (t_1 * (Math.sqrt(d) * Math.sqrt((1.0 / l))));
	} else if (d <= 2.6e+200) {
		tmp = t_3 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} else if (d <= 3.5e+303) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = (t_1 * Math.pow((d / l), 0.5)) * (1.0 - Math.pow((t_2 * (Math.sqrt(h) / Math.sqrt(l))), 2.0));
	}
	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 / math.sqrt((l / d))
	t_1 = math.pow((d / h), 0.5)
	t_2 = (M / ((d * 2.0) / D)) * math.sqrt(0.5)
	t_3 = 1.0 - math.pow((math.sqrt((h / l)) * t_2), 2.0)
	t_4 = math.sqrt(-d)
	tmp = 0
	if d <= -1e-105:
		tmp = ((t_4 / math.sqrt(-h)) * t_0) * t_3
	elif d <= 0.0:
		tmp = t_3 * (t_1 * (1.0 / (math.sqrt(-l) / t_4)))
	elif d <= 1e-50:
		tmp = t_3 * (t_1 * (math.sqrt(d) * math.sqrt((1.0 / l))))
	elif d <= 2.6e+200:
		tmp = t_3 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	elif d <= 3.5e+303:
		tmp = d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = (t_1 * math.pow((d / l), 0.5)) * (1.0 - math.pow((t_2 * (math.sqrt(h) / math.sqrt(l))), 2.0))
	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 / sqrt(Float64(l / d)))
	t_1 = Float64(d / h) ^ 0.5
	t_2 = Float64(Float64(M / Float64(Float64(d * 2.0) / D)) * sqrt(0.5))
	t_3 = Float64(1.0 - (Float64(sqrt(Float64(h / l)) * t_2) ^ 2.0))
	t_4 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1e-105)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-h))) * t_0) * t_3);
	elseif (d <= 0.0)
		tmp = Float64(t_3 * Float64(t_1 * Float64(1.0 / Float64(sqrt(Float64(-l)) / t_4))));
	elseif (d <= 1e-50)
		tmp = Float64(t_3 * Float64(t_1 * Float64(sqrt(d) * sqrt(Float64(1.0 / l)))));
	elseif (d <= 2.6e+200)
		tmp = Float64(t_3 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	elseif (d <= 3.5e+303)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(Float64(t_1 * (Float64(d / l) ^ 0.5)) * Float64(1.0 - (Float64(t_2 * Float64(sqrt(h) / sqrt(l))) ^ 2.0)));
	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 / sqrt((l / d));
	t_1 = (d / h) ^ 0.5;
	t_2 = (M / ((d * 2.0) / D)) * sqrt(0.5);
	t_3 = 1.0 - ((sqrt((h / l)) * t_2) ^ 2.0);
	t_4 = sqrt(-d);
	tmp = 0.0;
	if (d <= -1e-105)
		tmp = ((t_4 / sqrt(-h)) * t_0) * t_3;
	elseif (d <= 0.0)
		tmp = t_3 * (t_1 * (1.0 / (sqrt(-l) / t_4)));
	elseif (d <= 1e-50)
		tmp = t_3 * (t_1 * (sqrt(d) * sqrt((1.0 / l))));
	elseif (d <= 2.6e+200)
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	elseif (d <= 3.5e+303)
		tmp = d * sqrt((1.0 / (h * l)));
	else
		tmp = (t_1 * ((d / l) ^ 0.5)) * (1.0 - ((t_2 * (sqrt(h) / sqrt(l))) ^ 2.0));
	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[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[(N[(M / N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1e-105], N[(N[(N[(t$95$4 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[d, 0.0], N[(t$95$3 * N[(t$95$1 * N[(1.0 / N[(N[Sqrt[(-l)], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-50], N[(t$95$3 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.6e+200], N[(t$95$3 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+303], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(t$95$2 * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -9.99999999999999965e-106

   1. Initial program 21.7

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

   2. Applied egg-rr 20.2

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

   3. Applied egg-rr 20.1

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

   4. Applied egg-rr 12.0

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

   if -9.99999999999999965e-106 < d < 0.0

   1. Initial program 36.4

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

   2. Applied egg-rr 36.7

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

   3. Applied egg-rr 37.0

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

   4. Applied egg-rr 28.7

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

   if 0.0 < d < 1.00000000000000001e-50

   1. Initial program 33.6

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

   2. Applied egg-rr 32.5

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

   3. Applied egg-rr 26.3

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

   if 1.00000000000000001e-50 < d < 2.6000000000000001e200

   1. Initial program 19.8

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

   2. Applied egg-rr 18.2

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

   3. Applied egg-rr 18.2

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

   4. Applied egg-rr 10.5

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

   if 2.6000000000000001e200 < d < 3.50000000000000015e303

   1. Initial program 32.6

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

   2. Applied egg-rr 30.9

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

   3. Applied egg-rr 30.2

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

   4. Taylor expanded in d around inf 14.7

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

   5. Simplified 14.7

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

   if 3.50000000000000015e303 < d

   1. Initial program 27.1

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

   2. Applied egg-rr 20.6

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

   3. Applied egg-rr 20.7

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\frac{\sqrt{h}}{\sqrt{\ell}}} \cdot \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-105}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 0:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}}\right)\\ \mathbf{elif}\;d \leq 10^{-50}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+200}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+303}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))