Henrywood and Agarwal, Equation (12)

Average Error: 26.8 → 19.5

Time: 16.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{d}{D \cdot \frac{M}{2}}\\ t_1 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_2 := 1 + \frac{{\left(d \cdot \frac{\frac{2}{D}}{M}\right)}^{-2} \cdot -0.5}{\frac{\ell}{h}}\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;\ell \leq -1.2023678231762978 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right) \cdot t_2\\ \mathbf{elif}\;\ell \leq 3.1202928822756546 \cdot 10^{-191}:\\ \;\;\;\;\left(t_1 \cdot t_3\right) \cdot \left(1 + \frac{\sqrt{0.5} \cdot \frac{1}{t_0}}{\ell} \cdot \frac{\sqrt{0.5} \cdot \frac{-1}{t_0}}{\frac{1}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.6985331578856946 \cdot 10^{+223}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;\ell \leq 5.899305990900873 \cdot 10^{+261}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)\right)\\ \end{array} \]

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 (/ d (* D (/ M 2.0))))
        (t_1 (pow (/ d l) 0.5))
        (t_2 (+ 1.0 (/ (* (pow (* d (/ (/ 2.0 D) M)) -2.0) -0.5) (/ l h))))
        (t_3 (pow (/ d h) 0.5)))
   (if (<= l -1.2023678231762978e-21)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_1) t_2)
     (if (<= l 3.1202928822756546e-191)
       (*
        (* t_1 t_3)
        (+
         1.0
         (*
          (/ (* (sqrt 0.5) (/ 1.0 t_0)) l)
          (/ (* (sqrt 0.5) (/ -1.0 t_0)) (/ 1.0 h)))))
       (if (<= l 1.6985331578856946e+223)
         (* t_2 (* t_1 (/ (sqrt d) (sqrt h))))
         (if (<= l 5.899305990900873e+261)
           (* t_2 (* t_3 (* (sqrt d) (pow (/ 1.0 l) 0.5))))
           (* t_2 (* t_1 (* (sqrt d) (pow (/ 1.0 h) 0.5))))))))))

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 = d / (D * (M / 2.0));
	double t_1 = pow((d / l), 0.5);
	double t_2 = 1.0 + ((pow((d * ((2.0 / D) / M)), -2.0) * -0.5) / (l / h));
	double t_3 = pow((d / h), 0.5);
	double tmp;
	if (l <= -1.2023678231762978e-21) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_1) * t_2;
	} else if (l <= 3.1202928822756546e-191) {
		tmp = (t_1 * t_3) * (1.0 + (((sqrt(0.5) * (1.0 / t_0)) / l) * ((sqrt(0.5) * (-1.0 / t_0)) / (1.0 / h))));
	} else if (l <= 1.6985331578856946e+223) {
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)));
	} else if (l <= 5.899305990900873e+261) {
		tmp = t_2 * (t_3 * (sqrt(d) * pow((1.0 / l), 0.5)));
	} else {
		tmp = t_2 * (t_1 * (sqrt(d) * pow((1.0 / h), 0.5)));
	}
	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) :: tmp
    t_0 = d / (d_1 * (m / 2.0d0))
    t_1 = (d / l) ** 0.5d0
    t_2 = 1.0d0 + ((((d * ((2.0d0 / d_1) / m)) ** (-2.0d0)) * (-0.5d0)) / (l / h))
    t_3 = (d / h) ** 0.5d0
    if (l <= (-1.2023678231762978d-21)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_1) * t_2
    else if (l <= 3.1202928822756546d-191) then
        tmp = (t_1 * t_3) * (1.0d0 + (((sqrt(0.5d0) * (1.0d0 / t_0)) / l) * ((sqrt(0.5d0) * ((-1.0d0) / t_0)) / (1.0d0 / h))))
    else if (l <= 1.6985331578856946d+223) then
        tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)))
    else if (l <= 5.899305990900873d+261) then
        tmp = t_2 * (t_3 * (sqrt(d) * ((1.0d0 / l) ** 0.5d0)))
    else
        tmp = t_2 * (t_1 * (sqrt(d) * ((1.0d0 / h) ** 0.5d0)))
    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 = d / (D * (M / 2.0));
	double t_1 = Math.pow((d / l), 0.5);
	double t_2 = 1.0 + ((Math.pow((d * ((2.0 / D) / M)), -2.0) * -0.5) / (l / h));
	double t_3 = Math.pow((d / h), 0.5);
	double tmp;
	if (l <= -1.2023678231762978e-21) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_1) * t_2;
	} else if (l <= 3.1202928822756546e-191) {
		tmp = (t_1 * t_3) * (1.0 + (((Math.sqrt(0.5) * (1.0 / t_0)) / l) * ((Math.sqrt(0.5) * (-1.0 / t_0)) / (1.0 / h))));
	} else if (l <= 1.6985331578856946e+223) {
		tmp = t_2 * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	} else if (l <= 5.899305990900873e+261) {
		tmp = t_2 * (t_3 * (Math.sqrt(d) * Math.pow((1.0 / l), 0.5)));
	} else {
		tmp = t_2 * (t_1 * (Math.sqrt(d) * Math.pow((1.0 / h), 0.5)));
	}
	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 = d / (D * (M / 2.0))
	t_1 = math.pow((d / l), 0.5)
	t_2 = 1.0 + ((math.pow((d * ((2.0 / D) / M)), -2.0) * -0.5) / (l / h))
	t_3 = math.pow((d / h), 0.5)
	tmp = 0
	if l <= -1.2023678231762978e-21:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_1) * t_2
	elif l <= 3.1202928822756546e-191:
		tmp = (t_1 * t_3) * (1.0 + (((math.sqrt(0.5) * (1.0 / t_0)) / l) * ((math.sqrt(0.5) * (-1.0 / t_0)) / (1.0 / h))))
	elif l <= 1.6985331578856946e+223:
		tmp = t_2 * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	elif l <= 5.899305990900873e+261:
		tmp = t_2 * (t_3 * (math.sqrt(d) * math.pow((1.0 / l), 0.5)))
	else:
		tmp = t_2 * (t_1 * (math.sqrt(d) * math.pow((1.0 / h), 0.5)))
	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(d / Float64(D * Float64(M / 2.0)))
	t_1 = Float64(d / l) ^ 0.5
	t_2 = Float64(1.0 + Float64(Float64((Float64(d * Float64(Float64(2.0 / D) / M)) ^ -2.0) * -0.5) / Float64(l / h)))
	t_3 = Float64(d / h) ^ 0.5
	tmp = 0.0
	if (l <= -1.2023678231762978e-21)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_1) * t_2);
	elseif (l <= 3.1202928822756546e-191)
		tmp = Float64(Float64(t_1 * t_3) * Float64(1.0 + Float64(Float64(Float64(sqrt(0.5) * Float64(1.0 / t_0)) / l) * Float64(Float64(sqrt(0.5) * Float64(-1.0 / t_0)) / Float64(1.0 / h)))));
	elseif (l <= 1.6985331578856946e+223)
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	elseif (l <= 5.899305990900873e+261)
		tmp = Float64(t_2 * Float64(t_3 * Float64(sqrt(d) * (Float64(1.0 / l) ^ 0.5))));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(d) * (Float64(1.0 / h) ^ 0.5))));
	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 = d / (D * (M / 2.0));
	t_1 = (d / l) ^ 0.5;
	t_2 = 1.0 + ((((d * ((2.0 / D) / M)) ^ -2.0) * -0.5) / (l / h));
	t_3 = (d / h) ^ 0.5;
	tmp = 0.0;
	if (l <= -1.2023678231762978e-21)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_1) * t_2;
	elseif (l <= 3.1202928822756546e-191)
		tmp = (t_1 * t_3) * (1.0 + (((sqrt(0.5) * (1.0 / t_0)) / l) * ((sqrt(0.5) * (-1.0 / t_0)) / (1.0 / h))));
	elseif (l <= 1.6985331578856946e+223)
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)));
	elseif (l <= 5.899305990900873e+261)
		tmp = t_2 * (t_3 * (sqrt(d) * ((1.0 / l) ^ 0.5)));
	else
		tmp = t_2 * (t_1 * (sqrt(d) * ((1.0 / h) ^ 0.5)));
	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[(d / N[(D * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(N[Power[N[(d * N[(N[(2.0 / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * -0.5), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l, -1.2023678231762978e-21], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 3.1202928822756546e-191], N[(N[(t$95$1 * t$95$3), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6985331578856946e+223], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.899305990900873e+261], N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] * N[Power[N[(1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 5 regimes

2. if l < -1.2023678231762978e-21

   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. Applied egg-rr 26.9

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

   3. Applied egg-rr 26.9

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

   4. Applied egg-rr 26.8

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

   5. Applied egg-rr 18.6

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

   if -1.2023678231762978e-21 < l < 3.12029288227565458e-191

   1. Initial program 29.6

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

   2. Applied egg-rr 29.7

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

   3. Applied egg-rr 29.7

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

   4. Applied egg-rr 29.0

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

   5. Applied egg-rr 24.1

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

   if 3.12029288227565458e-191 < l < 1.69853315788569463e223

   1. Initial program 23.5

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

   2. Applied egg-rr 23.6

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

   3. Applied egg-rr 23.5

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

   4. Applied egg-rr 23.3

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

   5. Applied egg-rr 16.3

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

   if 1.69853315788569463e223 < l < 5.8993059909008734e261

   1. Initial program 31.8

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

   2. Applied egg-rr 31.8

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

   3. Applied egg-rr 31.8

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

   4. Applied egg-rr 31.6

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

   5. Applied egg-rr 24.1

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

   if 5.8993059909008734e261 < l

   1. Initial program 33.1

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

   2. Applied egg-rr 33.1

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

   3. Applied egg-rr 33.1

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

   4. Applied egg-rr 33.0

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

   5. Applied egg-rr 24.6

      \[\leadsto \left(\color{blue}{\left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5 \cdot {\left(\frac{\frac{2}{D}}{M} \cdot d\right)}^{-2}}{\frac{\ell}{h}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 19.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2023678231762978 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{{\left(d \cdot \frac{\frac{2}{D}}{M}\right)}^{-2} \cdot -0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 3.1202928822756546 \cdot 10^{-191}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 + \frac{\sqrt{0.5} \cdot \frac{1}{\frac{d}{D \cdot \frac{M}{2}}}}{\ell} \cdot \frac{\sqrt{0.5} \cdot \frac{-1}{\frac{d}{D \cdot \frac{M}{2}}}}{\frac{1}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.6985331578856946 \cdot 10^{+223}:\\ \;\;\;\;\left(1 + \frac{{\left(d \cdot \frac{\frac{2}{D}}{M}\right)}^{-2} \cdot -0.5}{\frac{\ell}{h}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;\ell \leq 5.899305990900873 \cdot 10^{+261}:\\ \;\;\;\;\left(1 + \frac{{\left(d \cdot \frac{\frac{2}{D}}{M}\right)}^{-2} \cdot -0.5}{\frac{\ell}{h}}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{\ell}\right)}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{{\left(d \cdot \frac{\frac{2}{D}}{M}\right)}^{-2} \cdot -0.5}{\frac{\ell}{h}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))))