Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.2% → 84.9%

Time: 17.9s

Precision: binary64

Cost: 14472

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-185}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{d}{{\left(M \cdot D\right)}^{2}}}}{d \cdot \left(\ell \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \]

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 (<= (* M D) 1e-185)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) h) l))))
   (if (<= (* M D) 5e+152)
     (* w0 (sqrt (- 1.0 (/ (/ h (/ d (pow (* M D) 2.0))) (* d (* l 4.0))))))
     (* w0 (sqrt (- 1.0 (* (pow (/ M (/ (* d 2.0) D)) 2.0) (/ h 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 ((M * D) <= 1e-185) {
		tmp = w0 * sqrt((1.0 - ((pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else if ((M * D) <= 5e+152) {
		tmp = w0 * sqrt((1.0 - ((h / (d / pow((M * D), 2.0))) / (d * (l * 4.0)))));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((M / ((d * 2.0) / D)), 2.0) * (h / 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 ((m * d) <= 1d-185) then
        tmp = w0 * sqrt((1.0d0 - ((((m * (0.5d0 * (d / d_1))) ** 2.0d0) * h) / l)))
    else if ((m * d) <= 5d+152) then
        tmp = w0 * sqrt((1.0d0 - ((h / (d_1 / ((m * d) ** 2.0d0))) / (d_1 * (l * 4.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - (((m / ((d_1 * 2.0d0) / d)) ** 2.0d0) * (h / 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 ((M * D) <= 1e-185) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else if ((M * D) <= 5e+152) {
		tmp = w0 * Math.sqrt((1.0 - ((h / (d / Math.pow((M * D), 2.0))) / (d * (l * 4.0)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((M / ((d * 2.0) / D)), 2.0) * (h / 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 (M * D) <= 1e-185:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((M * (0.5 * (D / d))), 2.0) * h) / l)))
	elif (M * D) <= 5e+152:
		tmp = w0 * math.sqrt((1.0 - ((h / (d / math.pow((M * D), 2.0))) / (d * (l * 4.0)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((M / ((d * 2.0) / D)), 2.0) * (h / 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(M * D) <= 1e-185)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * h) / l))));
	elseif (Float64(M * D) <= 5e+152)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / Float64(d / (Float64(M * D) ^ 2.0))) / Float64(d * Float64(l * 4.0))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(M / Float64(Float64(d * 2.0) / D)) ^ 2.0) * Float64(h / 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 ((M * D) <= 1e-185)
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 * (D / d))) ^ 2.0) * h) / l)));
	elseif ((M * D) <= 5e+152)
		tmp = w0 * sqrt((1.0 - ((h / (d / ((M * D) ^ 2.0))) / (d * (l * 4.0)))));
	else
		tmp = w0 * sqrt((1.0 - (((M / ((d * 2.0) / D)) ^ 2.0) * (h / 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[(M * D), $MachinePrecision], 1e-185], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 5e+152], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / N[(d / N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(M / N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.9999999999999999e-186

   1. Initial program 80.5%

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

   2. Simplified 80.7%

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

      [Start] 80.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 80.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 88.0%

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

      [Start] 80.7	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 88.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}} \]
      div-inv [=>] 88.0	\[ w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}} \]
      metadata-eval [=>] 88.0	\[ w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}} \]
      associate-*l* [=>] 88.0	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot h}{\ell}} \]

   if 9.9999999999999999e-186 < (*.f64 M D) < 5e152

   1. Initial program 81.5%

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

   2. Simplified 80.0%

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

      [Start] 81.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 81.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 80.0	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{d} \cdot \frac{D}{2}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 46.7%

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

   4. Simplified 76.7%

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

      [Start] 46.7	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
      associate-*r/ [=>] 46.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}} \]
      times-frac [=>] 46.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}} \]
      associate-*r* [=>] 49.5	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}} \]
      *-commutative [=>] 49.5	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2}}} \]
      unpow2 [=>] 49.5	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right)}{{d}^{2}}} \]
      unpow2 [=>] 49.5	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{h \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}}} \]
      unswap-sqr [=>] 76.7	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{h \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2}}} \]
      unpow2 [=>] 76.7	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{\color{blue}{d \cdot d}}} \]

   5. Applied egg-rr 86.6%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{h}{\frac{d}{{\left(D \cdot M\right)}^{2}}}}{\left(\ell \cdot 4\right) \cdot d}}} \]
      Proof

      [Start] 76.7	\[ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot d}} \]
      clear-num [=>] 76.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{0.25}}} \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot d}} \]
      associate-/r* [=>] 81.2	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{0.25}} \cdot \color{blue}{\frac{\frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d}}{d}}} \]
      frac-times [=>] 83.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d}}{\frac{\ell}{0.25} \cdot d}}} \]
      *-un-lft-identity [<=] 83.6	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d}}}{\frac{\ell}{0.25} \cdot d}} \]
      associate-/l* [=>] 86.6	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{h}{\frac{d}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}}}{\frac{\ell}{0.25} \cdot d}} \]
      pow2 [=>] 86.6	\[ w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{d}{\color{blue}{{\left(D \cdot M\right)}^{2}}}}}{\frac{\ell}{0.25} \cdot d}} \]
      div-inv [=>] 86.6	\[ w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{d}{{\left(D \cdot M\right)}^{2}}}}{\color{blue}{\left(\ell \cdot \frac{1}{0.25}\right)} \cdot d}} \]
      metadata-eval [=>] 86.6	\[ w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{d}{{\left(D \cdot M\right)}^{2}}}}{\left(\ell \cdot \color{blue}{4}\right) \cdot d}} \]

   if 5e152 < (*.f64 M D)

   1. Initial program 44.1%

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

   2. Simplified 48.7%

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

      [Start] 44.1	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-/l* [=>] 48.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-185}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{d}{{\left(M \cdot D\right)}^{2}}}}{d \cdot \left(\ell \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.9%
Cost	14344

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.25}{\ell}}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-268}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Accuracy	84.9%
Cost	14344

\[\begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-185}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+182}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{h}{d}\right)}{d \cdot \left(\ell \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \end{array} \]

Alternative 3
Accuracy	84.1%
Cost	14088

\[\begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;D \leq 1.2 \cdot 10^{+208}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{h}{d}\right)}{d \cdot \left(\ell \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	84.3%
Cost	8009

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq -1.85 \cdot 10^{+165} \lor \neg \left(h \leq 2 \cdot 10^{-198}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(h \cdot t_0\right) \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(t_0 \cdot \frac{M \cdot \left(D \cdot h\right)}{d}\right) \cdot \frac{-0.25}{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-268}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6
Accuracy	78.8%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023136 
(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))))))