Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 86.8%

Time: 26.0s

Precision: binary64

Cost: 21188

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d}}{\ell}}\\ \end{array} \]

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
 (if (<= (- 1.0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l))) 2e+292)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (* w0 (sqrt (- 1.0 (/ (* (/ 0.25 d) (/ (* h (* (* M D) (* M D))) d)) l))))))

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 tmp;
	if ((1.0 - (pow(((M * D) / (d * 2.0)), 2.0) * (h / l))) <= 2e+292) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (((0.25 / d) * ((h * ((M * D) * (M * D))) / d)) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((1.0d0 - ((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))) <= 2d+292) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (((0.25d0 / d_1) * ((h * ((m * d) * (m * d))) / d_1)) / l)))
    end if
    code = tmp
end function

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 tmp;
	if ((1.0 - (Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))) <= 2e+292) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((0.25 / d) * ((h * ((M * D) * (M * D))) / d)) / l)));
	}
	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):
	tmp = 0
	if (1.0 - (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))) <= 2e+292:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((0.25 / d) * ((h * ((M * D) * (M * D))) / d)) / l)))
	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)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))) <= 2e+292)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.25 / d) * Float64(Float64(h * Float64(Float64(M * D) * Float64(M * D))) / d)) / l))));
	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)
	tmp = 0.0;
	if ((1.0 - ((((M * D) / (d * 2.0)) ^ 2.0) * (h / l))) <= 2e+292)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (((0.25 / d) * ((h * ((M * D) * (M * D))) / d)) / l)));
	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_] := If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+292], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.25 / d), $MachinePrecision] * N[(N[(h * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 2e292

   1. Initial program 99.3%

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

   2. Simplified 99.3%

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

      [Start] 99.3%	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 99.3%	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 99.3%	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{d} \cdot \frac{D}{2}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   if 2e292 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 39.5%

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

   2. Simplified 41.6%

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

      [Start] 39.5%	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 41.6%	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 60.0%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}} \]
      Step-by-step derivation

      [Start] 41.6%	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      frac-times [=>] 39.5%	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 57.9%	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      *-commutative [=>] 57.9%	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}} \]
      frac-times [<=] 60.0%	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\frac{M}{d} \cdot \frac{D}{2}\right)}}^{2} \cdot h}{\ell}} \]
      div-inv [=>] 60.0%	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{d} \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}} \]
      metadata-eval [=>] 60.0%	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{d} \cdot \left(D \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}} \]

   4. Taylor expanded in M around 0 47.4%

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

   5. Simplified 56.5%

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

      [Start] 47.4%	\[ w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
      associate-*r/ [=>] 47.4%	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}}}{\ell}} \]
      associate-*r* [=>] 50.7%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}}}{\ell}} \]
      unpow2 [=>] 50.7%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}}}{\ell}} \]
      unpow2 [=>] 50.7%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}}}{\ell}} \]
      unswap-sqr [=>] 56.5%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}}}{\ell}} \]
      unpow2 [=>] 56.5%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h\right)}{\color{blue}{d \cdot d}}}{\ell}} \]

   6. Applied egg-rr 61.0%

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

      [Start] 56.5%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h\right)}{d \cdot d}}{\ell}} \]
      times-frac [=>] 61.0%	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d}}}{\ell}} \]
      pow2 [=>] 61.0%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{d}}{\ell}} \]

   7. Applied egg-rr 61.0%

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

      [Start] 61.0%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d}}{\ell}} \]
      unpow2 [=>] 61.0%	\[ w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{d}}{\ell}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.8%
Cost	21188

\[\begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d}}{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	82.2%
Cost	13956

\[\begin{array}{l} \mathbf{if}\;M \leq -8.5 \cdot 10^{-48}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d}}{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	85.9%
Cost	13824

\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}} \]

Alternative 4
Accuracy	76.6%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	81.0%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+188}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{d} \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d}}{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	71.9%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;M \leq -3.7 \cdot 10^{+147} \lor \neg \left(M \leq 3.3 \cdot 10^{-80}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7
Accuracy	72.3%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;M \leq -2.85 \cdot 10^{+147}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.1 \cdot 10^{-202}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{h}{d}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	72.9%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+151}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{M}}{M \cdot h} \cdot \frac{\ell}{D}}\right)\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{-203}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{h}{d}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	74.2%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\left(\ell \cdot \frac{d}{M}\right) \cdot \frac{d}{M \cdot h}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10
Accuracy	69.0%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;M \leq -1.12 \cdot 10^{+254} \lor \neg \left(M \leq 0.11\right):\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11
Accuracy	68.9%
Cost	1480

\[\begin{array}{l} t_0 := \frac{D \cdot D}{d}\\ \mathbf{if}\;M \leq -1.1 \cdot 10^{+252}:\\ \;\;\;\;-0.125 \cdot \left(t_0 \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \ell}\right)\\ \mathbf{elif}\;M \leq 0.11:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{d} \cdot \left(t_0 \cdot \frac{w0}{\frac{\frac{\ell}{h}}{M \cdot M}}\right)\\ \end{array} \]

Alternative 12
Accuracy	69.1%
Cost	1480

\[\begin{array}{l} \mathbf{if}\;M \leq -2.5 \cdot 10^{+248}:\\ \;\;\;\;\frac{-0.125}{d} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{d \cdot \ell}\\ \mathbf{elif}\;M \leq 0.11:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{d} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{w0}{\frac{\frac{\ell}{h}}{M \cdot M}}\right)\\ \end{array} \]

Alternative 13
Accuracy	67.6%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023271 
(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))))))