Henrywood and Agarwal, Equation (12)

Average Error: 26.3 → 19.3

Time: 50.7s

Precision: binary64

Cost: 104464

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := \left(t_1 \cdot t_2\right) \cdot t_0\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\left(\left({2}^{0.5} \cdot {\left(d \cdot \frac{0.5}{h}\right)}^{0.5}\right) \cdot t_2\right) \cdot t_0\\ \mathbf{elif}\;t_3 \leq 10^{-306}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

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) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (pow (/ d h) 0.5))
        (t_2 (pow (/ d l) 0.5))
        (t_3 (* (* t_1 t_2) t_0)))
   (if (<= t_3 (- INFINITY))
     (* (/ (pow (* D M) 2.0) d) (* (sqrt (/ h (pow l 3.0))) -0.125))
     (if (<= t_3 -2e-128)
       (* (* (* (pow 2.0 0.5) (pow (* d (/ 0.5 h)) 0.5)) t_2) t_0)
       (if (<= t_3 1e-306)
         (*
          (* d (- (sqrt (/ (/ 1.0 h) l))))
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) d) 2.0) 0.125))))
         (if (<= t_3 5e+273)
           (*
            (* t_2 t_1)
            (- 1.0 (* (/ h l) (* (pow (* D (/ M d)) 2.0) 0.125))))
           (* d (sqrt (/ (/ 1.0 l) 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) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = pow((d / h), 0.5);
	double t_2 = pow((d / l), 0.5);
	double t_3 = (t_1 * t_2) * t_0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (pow((D * M), 2.0) / d) * (sqrt((h / pow(l, 3.0))) * -0.125);
	} else if (t_3 <= -2e-128) {
		tmp = ((pow(2.0, 0.5) * pow((d * (0.5 / h)), 0.5)) * t_2) * t_0;
	} else if (t_3 <= 1e-306) {
		tmp = (d * -sqrt(((1.0 / h) / l))) * (1.0 - ((h / l) * (pow(((D * M) / d), 2.0) * 0.125)));
	} else if (t_3 <= 5e+273) {
		tmp = (t_2 * t_1) * (1.0 - ((h / l) * (pow((D * (M / d)), 2.0) * 0.125)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	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 = 1.0 - ((0.5 * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = Math.pow((d / h), 0.5);
	double t_2 = Math.pow((d / l), 0.5);
	double t_3 = (t_1 * t_2) * t_0;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.pow((D * M), 2.0) / d) * (Math.sqrt((h / Math.pow(l, 3.0))) * -0.125);
	} else if (t_3 <= -2e-128) {
		tmp = ((Math.pow(2.0, 0.5) * Math.pow((d * (0.5 / h)), 0.5)) * t_2) * t_0;
	} else if (t_3 <= 1e-306) {
		tmp = (d * -Math.sqrt(((1.0 / h) / l))) * (1.0 - ((h / l) * (Math.pow(((D * M) / d), 2.0) * 0.125)));
	} else if (t_3 <= 5e+273) {
		tmp = (t_2 * t_1) * (1.0 - ((h / l) * (Math.pow((D * (M / d)), 2.0) * 0.125)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / 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) / (2.0 * d)), 2.0)) * (h / l))
	t_1 = math.pow((d / h), 0.5)
	t_2 = math.pow((d / l), 0.5)
	t_3 = (t_1 * t_2) * t_0
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (math.pow((D * M), 2.0) / d) * (math.sqrt((h / math.pow(l, 3.0))) * -0.125)
	elif t_3 <= -2e-128:
		tmp = ((math.pow(2.0, 0.5) * math.pow((d * (0.5 / h)), 0.5)) * t_2) * t_0
	elif t_3 <= 1e-306:
		tmp = (d * -math.sqrt(((1.0 / h) / l))) * (1.0 - ((h / l) * (math.pow(((D * M) / d), 2.0) * 0.125)))
	elif t_3 <= 5e+273:
		tmp = (t_2 * t_1) * (1.0 - ((h / l) * (math.pow((D * (M / d)), 2.0) * 0.125)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / 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(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(d / h) ^ 0.5
	t_2 = Float64(d / l) ^ 0.5
	t_3 = Float64(Float64(t_1 * t_2) * t_0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64((Float64(D * M) ^ 2.0) / d) * Float64(sqrt(Float64(h / (l ^ 3.0))) * -0.125));
	elseif (t_3 <= -2e-128)
		tmp = Float64(Float64(Float64((2.0 ^ 0.5) * (Float64(d * Float64(0.5 / h)) ^ 0.5)) * t_2) * t_0);
	elseif (t_3 <= 1e-306)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l)))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / d) ^ 2.0) * 0.125))));
	elseif (t_3 <= 5e+273)
		tmp = Float64(Float64(t_2 * t_1) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(D * Float64(M / d)) ^ 2.0) * 0.125))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / 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) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_1 = (d / h) ^ 0.5;
	t_2 = (d / l) ^ 0.5;
	t_3 = (t_1 * t_2) * t_0;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (((D * M) ^ 2.0) / d) * (sqrt((h / (l ^ 3.0))) * -0.125);
	elseif (t_3 <= -2e-128)
		tmp = (((2.0 ^ 0.5) * ((d * (0.5 / h)) ^ 0.5)) * t_2) * t_0;
	elseif (t_3 <= 1e-306)
		tmp = (d * -sqrt(((1.0 / h) / l))) * (1.0 - ((h / l) * ((((D * M) / d) ^ 2.0) * 0.125)));
	elseif (t_3 <= 5e+273)
		tmp = (t_2 * t_1) * (1.0 - ((h / l) * (((D * (M / d)) ^ 2.0) * 0.125)));
	else
		tmp = d * sqrt(((1.0 / l) / 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[(2.0 * d), $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[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-128], N[(N[(N[(N[Power[2.0, 0.5], $MachinePrecision] * N[Power[N[(d * N[(0.5 / h), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 1e-306], N[(N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+273], N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -inf.0

   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. Taylor expanded in d around 0 59.8

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

   3. Simplified 57.1

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

      [Start] 59.8	\[ -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      rational.json-simplify-43 [=>] 59.8	\[ \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)} \]
      exponential.json-simplify-27 [=>] 57.1	\[ \frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right) \]

   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)))) < -2.00000000000000011e-128

   1. Initial program 1.6

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

   2. Applied egg-rr 1.7

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

   if -2.00000000000000011e-128 < (*.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)))) < 1.00000000000000003e-306

   1. Initial program 37.8

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

   2. Simplified 42.4

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

      [Start] 37.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      exponential.json-simplify-27 [=>] 41.5	\[ \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 41.5	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 41.5	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      rational.json-simplify-49 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(0.5 \cdot {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      rational.json-simplify-2 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(0.5 \cdot {\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 42.4

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 0\right)}\right) \]

   4. Simplified 42.4

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)}\right) \]
      Proof

      [Start] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 0\right)\right) \]
      rational.json-simplify-4 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      rational.json-simplify-2 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right)}\right) \]
      metadata-eval [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\frac{2}{4}}\right)\right) \]
      rational.json-simplify-49 [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{2 \cdot \frac{h}{\ell}}{4}}\right) \]
      metadata-eval [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{2 \cdot \frac{h}{\ell}}{\color{blue}{\frac{2}{0.5}}}\right) \]
      rational.json-simplify-61 [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{2}{2 \cdot \frac{h}{\ell}}}}\right) \]
      rational.json-simplify-49 [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2}{2 \cdot \frac{h}{\ell}}}}\right) \]
      rational.json-simplify-61 [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{2 \cdot \frac{h}{\ell}}{\frac{2}{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      rational.json-simplify-49 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \frac{2}{\frac{2}{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      rational.json-simplify-46 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{2}{\color{blue}{\frac{\frac{2}{0.5}}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      metadata-eval [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{2}{\frac{\color{blue}{4}}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}\right) \]
      rational.json-simplify-61 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{4}{2}}}\right) \]
      metadata-eval [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\color{blue}{2}}\right) \]
      rational.json-simplify-43 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(\frac{M}{d} \cdot D\right)\right)}}^{2}}{2}\right) \]
      rational.json-simplify-2 [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)}^{2}}{2}\right) \]
      exponential.json-simplify-26 [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{{0.5}^{2} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{2}\right) \]
      metadata-eval [=>] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{0.25} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{2}\right) \]
      metadata-eval [<=] 42.4	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{\frac{2}{8}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{2}\right) \]

   5. Applied egg-rr 42.6

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\frac{h}{\frac{d}{\ell}}}\right)}^{0.5} + 0\right)} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]

   6. Simplified 42.4

      \[\leadsto \color{blue}{{\left(\frac{\frac{d}{\ell}}{\frac{h}{d}}\right)}^{0.5}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      Proof

      [Start] 42.6	\[ \left({\left(\frac{d}{\frac{h}{\frac{d}{\ell}}}\right)}^{0.5} + 0\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-4 [=>] 42.6	\[ \color{blue}{{\left(\frac{d}{\frac{h}{\frac{d}{\ell}}}\right)}^{0.5}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-61 [=>] 42.4	\[ {\color{blue}{\left(\frac{\frac{d}{\ell}}{\frac{h}{d}}\right)}}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]

   7. Taylor expanded in D around 0 41.5

      \[\leadsto {\left(\frac{\frac{d}{\ell}}{\frac{h}{d}}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2} \cdot 0.125\right)\right) \]

   8. Taylor expanded in d around -inf 26.9

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

   9. Simplified 26.8

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

      [Start] 26.9	\[ \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-43 [=>] 26.9	\[ \color{blue}{\left(d \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot -1\right)\right)} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-9 [=>] 26.9	\[ \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-46 [=>] 26.8	\[ \left(d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right) \]
      rational.json-simplify-44 [=>] 26.8	\[ \left(d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right) \]

   if 1.00000000000000003e-306 < (*.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)))) < 4.99999999999999961e273

   1. Initial program 1.1

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

   2. Simplified 15.9

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

      [Start] 1.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      exponential.json-simplify-27 [=>] 15.8	\[ \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 15.8	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 15.8	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      rational.json-simplify-49 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(0.5 \cdot {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      rational.json-simplify-2 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left(0.5 \cdot {\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 15.9

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 0\right)}\right) \]

   4. Simplified 15.9

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)}\right) \]
      Proof

      [Start] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 0\right)\right) \]
      rational.json-simplify-4 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      rational.json-simplify-2 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right)}\right) \]
      metadata-eval [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\frac{2}{4}}\right)\right) \]
      rational.json-simplify-49 [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{2 \cdot \frac{h}{\ell}}{4}}\right) \]
      metadata-eval [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{2 \cdot \frac{h}{\ell}}{\color{blue}{\frac{2}{0.5}}}\right) \]
      rational.json-simplify-61 [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{2}{2 \cdot \frac{h}{\ell}}}}\right) \]
      rational.json-simplify-49 [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2}{2 \cdot \frac{h}{\ell}}}}\right) \]
      rational.json-simplify-61 [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{2 \cdot \frac{h}{\ell}}{\frac{2}{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      rational.json-simplify-49 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \frac{2}{\frac{2}{0.5 \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      rational.json-simplify-46 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{2}{\color{blue}{\frac{\frac{2}{0.5}}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}}\right) \]
      metadata-eval [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{2}{\frac{\color{blue}{4}}{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}}\right) \]
      rational.json-simplify-61 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{4}{2}}}\right) \]
      metadata-eval [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\color{blue}{2}}\right) \]
      rational.json-simplify-43 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(\frac{M}{d} \cdot D\right)\right)}}^{2}}{2}\right) \]
      rational.json-simplify-2 [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{{\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)}^{2}}{2}\right) \]
      exponential.json-simplify-26 [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{{0.5}^{2} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{2}\right) \]
      metadata-eval [=>] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{0.25} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{2}\right) \]
      metadata-eval [<=] 15.9	\[ {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \frac{\color{blue}{\frac{2}{8}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{2}\right) \]

   5. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right) \]

   if 4.99999999999999961e273 < (*.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 63.0

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

   2. Taylor expanded in d around inf 43.3

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

   3. Simplified 43.3

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

      [Start] 43.3	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      rational.json-simplify-2 [=>] 43.3	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      rational.json-simplify-46 [=>] 43.3	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 19.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\left(\left({2}^{0.5} \cdot {\left(d \cdot \frac{0.5}{h}\right)}^{0.5}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-306}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.3
Cost	104464

\[\begin{array}{l} t_0 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := \left(t_1 \cdot t_2\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;\left({\left(d \cdot \frac{2}{\ell}\right)}^{0.5} \cdot {\left(d \cdot \frac{0.5}{h}\right)}^{0.5}\right) \cdot t_0\\ \mathbf{elif}\;t_3 \leq 10^{-306}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot t_0\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 2
Error	21.6
Cost	21528

\[\begin{array}{l} t_0 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ t_1 := t_0 \cdot 0.125\\ t_2 := \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot t_1\right)\\ \mathbf{if}\;h \leq -4.6 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -9.5 \cdot 10^{-307}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{elif}\;h \leq 2.45 \cdot 10^{+169}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{+198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+269}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(1 - \frac{t_0}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.85 \cdot 10^{+296}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\frac{\ell}{t_1}}\right)\\ \end{array} \]

Alternative 3
Error	23.5
Cost	21264

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\\ t_2 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+216}:\\ \;\;\;\;{\left(d \cdot \frac{\frac{d}{\ell}}{h}\right)}^{0.5} \cdot t_1\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-122}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-293}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_2 \cdot 0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(1 - \frac{t_2}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1\\ \end{array} \]

Alternative 4
Error	24.5
Cost	15064

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_1 := d \cdot t_0\\ t_2 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\\ t_3 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ t_4 := {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_3 \cdot 0.125\right)\right)\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+206}:\\ \;\;\;\;{\left(d \cdot \frac{\frac{d}{\ell}}{h}\right)}^{0.5} \cdot t_2\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-282}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot t_2\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+134}:\\ \;\;\;\;d \cdot \left(t_0 \cdot \left(1 - \frac{t_3}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+200}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+284}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.8
Cost	14932

\[\begin{array}{l} t_0 := {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_2 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\\ t_3 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+206}:\\ \;\;\;\;{\left(d \cdot \frac{\frac{d}{\ell}}{h}\right)}^{0.5} \cdot t_2\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-282}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot t_2\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;d \cdot \left(t_1 \cdot \left(1 - \frac{t_3}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+199}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_3 \cdot 0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+276}:\\ \;\;\;\;d \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_2\\ \end{array} \]

Alternative 6
Error	23.3
Cost	14932

\[\begin{array}{l} t_0 := {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\\ t_2 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_3 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+217}:\\ \;\;\;\;{\left(d \cdot \frac{\frac{d}{\ell}}{h}\right)}^{0.5} \cdot t_1\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+135}:\\ \;\;\;\;d \cdot \left(t_2 \cdot \left(1 - \frac{t_3}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+199}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_3 \cdot 0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+273}:\\ \;\;\;\;d \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1\\ \end{array} \]

Alternative 7
Error	23.5
Cost	14800

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_1 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ t_2 := \frac{t_1}{\ell}\\ t_3 := {\left(d \cdot \frac{\frac{d}{\ell}}{h}\right)}^{0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_1 \cdot 0.125\right)\right)\\ \mathbf{if}\;h \leq -6.2 \cdot 10^{+97}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -9.5 \cdot 10^{-307}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+169}:\\ \;\;\;\;t_0 \cdot \left(d \cdot \left(1 - h \cdot \left(0.125 \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.1 \cdot 10^{+196}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(t_0 \cdot \left(1 - t_2 \cdot \left(h \cdot 0.125\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	23.8
Cost	14404

\[\begin{array}{l} t_0 := {\left(D \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;h \leq -9.5 \cdot 10^{-307}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(t_0 \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(1 - \frac{t_0}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	24.5
Cost	14404

\[\begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-235}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(1 - \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	25.5
Cost	14340

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq 2.15 \cdot 10^{-235}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(t_0 \cdot \left(1 - \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot 0.125\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	28.1
Cost	7044

\[\begin{array}{l} \mathbf{if}\;d \leq 9.8 \cdot 10^{-203}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 12
Error	28.0
Cost	7044

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq 9.8 \cdot 10^{-203}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 13
Error	43.1
Cost	6848

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

Reproduce

herbie shell --seed 2023073 
(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)))))