Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 20.1

Time: 1.6min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_1 := \sqrt{\frac{\ell}{h}}\\ t_2 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_4 := {\left(\frac{d}{h}\right)}^{0.5} \cdot t_2\\ t_5 := t_4 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_6 := 0.5 \cdot \frac{M \cdot D}{d}\\ \mathbf{if}\;t_5 \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\left(t_2 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{t_6}{t_1}\right)}^{2}\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t_4 \cdot \left(1 - {\left(\frac{t_6 \cdot \sqrt{0.5}}{t_1}\right)}^{2}\right)\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(1 + \frac{\left(\left(M \cdot M\right) \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot d}\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

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 (pow (/ d h) 0.25))
        (t_1 (sqrt (/ l h)))
        (t_2 (pow (/ d l) 0.5))
        (t_3 (* d (sqrt (/ 1.0 (* h l)))))
        (t_4 (* (pow (/ d h) 0.5) t_2))
        (t_5
         (* t_4 (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_6 (* 0.5 (/ (* M D) d))))
   (if (<= t_5 -1e-247)
     (* (* t_2 (* t_0 t_0)) (- 1.0 (* 0.5 (pow (/ t_6 t_1) 2.0))))
     (if (<= t_5 0.0)
       t_3
       (if (<= t_5 2e+290)
         (* t_4 (- 1.0 (pow (/ (* t_6 (sqrt 0.5)) t_1) 2.0)))
         (if (<= t_5 INFINITY)
           t_3
           (*
            t_4
            (+
             1.0
             (/ (* (* (* M M) (/ (* D (* h D)) (* d d))) -0.125) l)))))))))

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 = pow((d / h), 0.25);
	double t_1 = sqrt((l / h));
	double t_2 = pow((d / l), 0.5);
	double t_3 = d * sqrt((1.0 / (h * l)));
	double t_4 = pow((d / h), 0.5) * t_2;
	double t_5 = t_4 * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_6 = 0.5 * ((M * D) / d);
	double tmp;
	if (t_5 <= -1e-247) {
		tmp = (t_2 * (t_0 * t_0)) * (1.0 - (0.5 * pow((t_6 / t_1), 2.0)));
	} else if (t_5 <= 0.0) {
		tmp = t_3;
	} else if (t_5 <= 2e+290) {
		tmp = t_4 * (1.0 - pow(((t_6 * sqrt(0.5)) / t_1), 2.0));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 + ((((M * M) * ((D * (h * D)) / (d * d))) * -0.125) / l));
	}
	return tmp;
}

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.pow((d / h), 0.25);
	double t_1 = Math.sqrt((l / h));
	double t_2 = Math.pow((d / l), 0.5);
	double t_3 = d * Math.sqrt((1.0 / (h * l)));
	double t_4 = Math.pow((d / h), 0.5) * t_2;
	double t_5 = t_4 * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_6 = 0.5 * ((M * D) / d);
	double tmp;
	if (t_5 <= -1e-247) {
		tmp = (t_2 * (t_0 * t_0)) * (1.0 - (0.5 * Math.pow((t_6 / t_1), 2.0)));
	} else if (t_5 <= 0.0) {
		tmp = t_3;
	} else if (t_5 <= 2e+290) {
		tmp = t_4 * (1.0 - Math.pow(((t_6 * Math.sqrt(0.5)) / t_1), 2.0));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = t_4 * (1.0 + ((((M * M) * ((D * (h * D)) / (d * d))) * -0.125) / l));
	}
	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.pow((d / h), 0.25)
	t_1 = math.sqrt((l / h))
	t_2 = math.pow((d / l), 0.5)
	t_3 = d * math.sqrt((1.0 / (h * l)))
	t_4 = math.pow((d / h), 0.5) * t_2
	t_5 = t_4 * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_6 = 0.5 * ((M * D) / d)
	tmp = 0
	if t_5 <= -1e-247:
		tmp = (t_2 * (t_0 * t_0)) * (1.0 - (0.5 * math.pow((t_6 / t_1), 2.0)))
	elif t_5 <= 0.0:
		tmp = t_3
	elif t_5 <= 2e+290:
		tmp = t_4 * (1.0 - math.pow(((t_6 * math.sqrt(0.5)) / t_1), 2.0))
	elif t_5 <= math.inf:
		tmp = t_3
	else:
		tmp = t_4 * (1.0 + ((((M * M) * ((D * (h * D)) / (d * d))) * -0.125) / l))
	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 / h) ^ 0.25
	t_1 = sqrt(Float64(l / h))
	t_2 = Float64(d / l) ^ 0.5
	t_3 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	t_4 = Float64((Float64(d / h) ^ 0.5) * t_2)
	t_5 = Float64(t_4 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_6 = Float64(0.5 * Float64(Float64(M * D) / d))
	tmp = 0.0
	if (t_5 <= -1e-247)
		tmp = Float64(Float64(t_2 * Float64(t_0 * t_0)) * Float64(1.0 - Float64(0.5 * (Float64(t_6 / t_1) ^ 2.0))));
	elseif (t_5 <= 0.0)
		tmp = t_3;
	elseif (t_5 <= 2e+290)
		tmp = Float64(t_4 * Float64(1.0 - (Float64(Float64(t_6 * sqrt(0.5)) / t_1) ^ 2.0)));
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(t_4 * Float64(1.0 + Float64(Float64(Float64(Float64(M * M) * Float64(Float64(D * Float64(h * D)) / Float64(d * d))) * -0.125) / l)));
	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 / h) ^ 0.25;
	t_1 = sqrt((l / h));
	t_2 = (d / l) ^ 0.5;
	t_3 = d * sqrt((1.0 / (h * l)));
	t_4 = ((d / h) ^ 0.5) * t_2;
	t_5 = t_4 * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_6 = 0.5 * ((M * D) / d);
	tmp = 0.0;
	if (t_5 <= -1e-247)
		tmp = (t_2 * (t_0 * t_0)) * (1.0 - (0.5 * ((t_6 / t_1) ^ 2.0)));
	elseif (t_5 <= 0.0)
		tmp = t_3;
	elseif (t_5 <= 2e+290)
		tmp = t_4 * (1.0 - (((t_6 * sqrt(0.5)) / t_1) ^ 2.0));
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = t_4 * (1.0 + ((((M * M) * ((D * (h * D)) / (d * d))) * -0.125) / l));
	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[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-247], N[(N[(t$95$2 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$6 / t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], t$95$3, If[LessEqual[t$95$5, 2e+290], N[(t$95$4 * N[(1.0 - N[Power[N[(N[(t$95$6 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$3, N[(t$95$4 * N[(1.0 + N[(N[(N[(N[(M * M), $MachinePrecision] * N[(N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -1e-247

   1. Initial program 29.2

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

   2. Applied egg-rr 31.2

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

   3. Applied egg-rr 29.4

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

   4. Applied egg-rr 29.4

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

   5. Applied egg-rr 21.0

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

   if -1e-247 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 2.00000000000000012e290 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 51.8

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

   2. Taylor expanded in d around inf 31.4

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

   3. Simplified 31.4

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.00000000000000012e290

   1. Initial program 0.9

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 0.9

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 64.0

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

   2. Applied egg-rr 57.3

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

   3. Taylor expanded in M around 0 57.7

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

   4. Simplified 56.7

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot d}\right)}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{0.5 \cdot \frac{M \cdot D}{d}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\frac{\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq \infty:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{\left(\left(M \cdot M\right) \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot d}\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022325 
(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)))))