Henrywood and Agarwal, Equation (9a)

Average Error: 21.34% → 13.73%

Time: 16.5s

Precision: binary64

Cost: 8264

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

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot \left(h \cdot \frac{D}{\ell \cdot \frac{\frac{d \cdot d}{D}}{M}}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -6 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(t_0 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D d) M)))
   (if (<= (/ h l) (- INFINITY))
     (*
      w0
      (sqrt (+ 1.0 (* (* M (* h (/ D (* l (/ (/ (* d d) D) M))))) -0.25))))
     (if (<= (/ h l) -6e-321)
       (* w0 (sqrt (+ 1.0 (* (* t_0 (* t_0 (/ h l))) -0.25))))
       w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * M;
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 + ((M * (h * (D / (l * (((d * d) / D) / M))))) * -0.25)));
	} else if ((h / l) <= -6e-321) {
		tmp = w0 * sqrt((1.0 + ((t_0 * (t_0 * (h / l))) * -0.25)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * M;
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 + ((M * (h * (D / (l * (((d * d) / D) / M))))) * -0.25)));
	} else if ((h / l) <= -6e-321) {
		tmp = w0 * Math.sqrt((1.0 + ((t_0 * (t_0 * (h / l))) * -0.25)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	t_0 = (D / d) * M
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 + ((M * (h * (D / (l * (((d * d) / D) / M))))) * -0.25)))
	elif (h / l) <= -6e-321:
		tmp = w0 * math.sqrt((1.0 + ((t_0 * (t_0 * (h / l))) * -0.25)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D / d) * M)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M * Float64(h * Float64(D / Float64(l * Float64(Float64(Float64(d * d) / D) / M))))) * -0.25))));
	elseif (Float64(h / l) <= -6e-321)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 * Float64(t_0 * Float64(h / l))) * -0.25))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (D / d) * M;
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 + ((M * (h * (D / (l * (((d * d) / D) / M))))) * -0.25)));
	elseif ((h / l) <= -6e-321)
		tmp = w0 * sqrt((1.0 + ((t_0 * (t_0 * (h / l))) * -0.25)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M * N[(h * N[(D / N[(l * N[(N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -6e-321], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 100

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 100

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 100	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 100	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 100	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in D around 0 68.03

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

   4. Simplified 51.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}}} \]
      Proof

      [Start] 68.03	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
      *-commutative [=>] 68.03	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell \cdot {d}^{2}}} \]
      *-commutative [=>] 68.03	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]
      *-commutative [<=] 68.03	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
      associate-*r* [=>] 64.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 64.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 64.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
      swap-sqr [<=] 55.97	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2} \cdot \ell}} \]
      associate-*l* [=>] 54.14	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 54.14	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]
      associate-*l* [=>] 51.8	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}} \]

   5. Applied egg-rr 62.1

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{D} \cdot \frac{d \cdot \ell}{M \cdot h}} \cdot M\right)}} \]

   6. Taylor expanded in D around 0 66.05

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\frac{{D}^{2} \cdot \left(h \cdot M\right)}{{d}^{2} \cdot \ell}} \cdot M\right)} \]

   7. Simplified 51.75

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(h \cdot \frac{D}{\ell \cdot \frac{\frac{d \cdot d}{D}}{M}}\right)} \cdot M\right)} \]
      Proof

      [Start] 66.05	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot M\right)}{{d}^{2} \cdot \ell} \cdot M\right)} \]
      *-commutative [=>] 66.05	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2} \cdot \color{blue}{\left(M \cdot h\right)}}{{d}^{2} \cdot \ell} \cdot M\right)} \]
      times-frac [=>] 73.43	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{M \cdot h}{\ell}\right)} \cdot M\right)} \]
      unpow2 [=>] 73.43	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{{D}^{2}}{\color{blue}{d \cdot d}} \cdot \frac{M \cdot h}{\ell}\right) \cdot M\right)} \]
      associate-/r* [=>] 70.43	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\color{blue}{\frac{\frac{{D}^{2}}{d}}{d}} \cdot \frac{M \cdot h}{\ell}\right) \cdot M\right)} \]
      unpow2 [=>] 70.43	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{d} \cdot \frac{M \cdot h}{\ell}\right) \cdot M\right)} \]
      associate-*r/ [<=] 67.37	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \frac{M \cdot h}{\ell}\right) \cdot M\right)} \]
      associate-*l/ [<=] 67.36	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D \cdot \frac{D}{d}}{d} \cdot \color{blue}{\left(\frac{M}{\ell} \cdot h\right)}\right) \cdot M\right)} \]
      associate-*r* [=>] 52.07	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\left(\frac{D \cdot \frac{D}{d}}{d} \cdot \frac{M}{\ell}\right) \cdot h\right)} \cdot M\right)} \]
      associate-/r/ [<=] 51.38	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\color{blue}{\frac{D \cdot \frac{D}{d}}{\frac{d}{\frac{M}{\ell}}}} \cdot h\right) \cdot M\right)} \]
      *-commutative [=>] 51.38	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(h \cdot \frac{D \cdot \frac{D}{d}}{\frac{d}{\frac{M}{\ell}}}\right)} \cdot M\right)} \]
      associate-/l* [=>] 51.16	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \color{blue}{\frac{D}{\frac{\frac{d}{\frac{M}{\ell}}}{\frac{D}{d}}}}\right) \cdot M\right)} \]
      associate-/l* [<=] 54.09	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\color{blue}{\frac{\frac{d}{\frac{M}{\ell}} \cdot d}{D}}}\right) \cdot M\right)} \]
      associate-*r/ [<=] 51.16	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\color{blue}{\frac{d}{\frac{M}{\ell}} \cdot \frac{d}{D}}}\right) \cdot M\right)} \]
      *-commutative [=>] 51.16	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\color{blue}{\frac{d}{D} \cdot \frac{d}{\frac{M}{\ell}}}}\right) \cdot M\right)} \]
      associate-/r/ [=>] 45.91	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{M} \cdot \ell\right)}}\right) \cdot M\right)} \]
      associate-*r* [=>] 45.89	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{M}\right) \cdot \ell}}\right) \cdot M\right)} \]
      *-commutative [=>] 45.89	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\color{blue}{\ell \cdot \left(\frac{d}{D} \cdot \frac{d}{M}\right)}}\right) \cdot M\right)} \]
      associate-*r/ [=>] 47.73	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\ell \cdot \color{blue}{\frac{\frac{d}{D} \cdot d}{M}}}\right) \cdot M\right)} \]
      associate-*l/ [=>] 51.75	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(h \cdot \frac{D}{\ell \cdot \frac{\color{blue}{\frac{d \cdot d}{D}}}{M}}\right) \cdot M\right)} \]

   if -inf.0 < (/.f64 h l) < -5.99796e-321

   1. Initial program 22.16

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 21.96

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 22.16	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 21.96	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 21.96	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in D around 0 57.55

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

   4. Simplified 37.61

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}}} \]
      Proof

      [Start] 57.55	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
      *-commutative [=>] 57.55	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell \cdot {d}^{2}}} \]
      *-commutative [=>] 57.55	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]
      *-commutative [<=] 57.55	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
      associate-*r* [=>] 56.28	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 56.28	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 56.28	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
      swap-sqr [<=] 43.64	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2} \cdot \ell}} \]
      associate-*l* [=>] 41.36	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
      unpow2 [=>] 41.36	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]
      associate-*l* [=>] 37.61	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}} \]

   5. Applied egg-rr 19.36

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{h}{\ell}\right)\right)}} \]

   if -5.99796e-321 < (/.f64 h l)

   1. Initial program 11.35

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 11.42

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 11.35	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 11.42	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 11.42	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in D around 0 3.75

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.73

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot \left(h \cdot \frac{D}{\ell \cdot \frac{\frac{d \cdot d}{D}}{M}}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -6 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{h}{\ell}\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	13.14%
Cost	7744

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ w0 \cdot \sqrt{1 - 0.25 \cdot \left(t_0 \cdot \frac{h}{\frac{\ell}{t_0}}\right)} \end{array} \]

Alternative 2
Error	21.3%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))