Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 87.6%

Time: 25.5s

Precision: binary64

Cost: 7872

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

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{D \cdot M}{d}\\ w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(t_0 \cdot h\right)}{\ell}} \end{array} \]

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
 (let* ((t_0 (* 0.5 (/ (* D M) d))))
   (* w0 (sqrt (- 1.0 (/ (* t_0 (* t_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 t_0 = 0.5 * ((D * M) / d);
	return w0 * sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
}

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) :: t_0
    t_0 = 0.5d0 * ((d * m) / d_1)
    code = w0 * sqrt((1.0d0 - ((t_0 * (t_0 * h)) / l)))
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 t_0 = 0.5 * ((D * M) / d);
	return w0 * Math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
}

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 = 0.5 * ((D * M) / d)
	return w0 * math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)))

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(0.5 * Float64(Float64(D * M) / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * h)) / l))))
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 = code(w0, M, D, h, l, d)
	t_0 = 0.5 * ((D * M) / d);
	tmp = w0 * sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
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[(0.5 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.3%

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

2. Applied egg-rr 85.6%

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

   [Start] 82.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   associate-*r/ [=>] 85.5	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
   clear-num [=>] 85.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}} \]
   div-inv [=>] 85.2	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2} \cdot h}}} \]
   associate-*l* [=>] 85.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\color{blue}{\left(M \cdot \left(D \cdot \frac{1}{2 \cdot d}\right)\right)}}^{2} \cdot h}}} \]
   associate-/r* [=>] 85.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)}^{2} \cdot h}}} \]
   metadata-eval [=>] 85.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{\color{blue}{0.5}}{d}\right)\right)}^{2} \cdot h}}} \]

3. Simplified 85.2%

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

   [Start] 85.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}}} \]
   associate-/r/ [=>] 85.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\ell} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}} \]
   *-commutative [=>] 85.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}} \]
   *-commutative [=>] 85.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)} \cdot \frac{1}{\ell}} \]
   *-commutative [=>] 85.6	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \color{blue}{\left(\frac{0.5}{d} \cdot D\right)}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   associate-/r/ [<=] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   metadata-eval [<=] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{\color{blue}{\frac{1}{2}}}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   associate-/r* [<=] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \color{blue}{\frac{1}{2 \cdot \frac{d}{D}}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   associate-*r/ [=>] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\color{blue}{\left(\frac{M \cdot 1}{2 \cdot \frac{d}{D}}\right)}}^{2}\right) \cdot \frac{1}{\ell}} \]
   *-rgt-identity [=>] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{\color{blue}{M}}{2 \cdot \frac{d}{D}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   associate-*r/ [=>] 86.0	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\color{blue}{\frac{2 \cdot d}{D}}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   associate-/r/ [=>] 85.2	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2}\right) \cdot \frac{1}{\ell}} \]
   *-commutative [<=] 85.2	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{1}{\ell}} \]

4. Applied egg-rr 84.7%

   \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\right)} - 1\right)} \]
   Step-by-step derivation

   [Start] 85.2	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
   expm1-log1p-u [=>] 84.7	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right) \cdot \frac{1}{\ell}}\right)\right)} \]
   expm1-udef [=>] 84.7	\[ w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right) \cdot \frac{1}{\ell}}\right)} - 1\right)} \]
   associate-*l* [=>] 84.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{h \cdot \left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\ell}\right)}}\right)} - 1\right) \]
   un-div-inv [=>] 84.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}}\right)} - 1\right) \]

5. Simplified 86.3%

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

   [Start] 84.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\right)} - 1\right) \]
   expm1-def [=>] 84.7	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\right)\right)} \]
   expm1-log1p [=>] 85.2	\[ w0 \cdot \color{blue}{\sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}} \]
   associate-*r/ [=>] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2}}{\ell}} \]
   *-lft-identity [<=] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{\color{blue}{1 \cdot \left(D \cdot M\right)}}{2 \cdot d}\right)}^{2}}{\ell}} \]
   associate-*l/ [<=] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{1}{2 \cdot d} \cdot \left(D \cdot M\right)\right)}}^{2}}{\ell}} \]
   associate-/r/ [<=] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}}^{2}}{\ell}} \]
   associate-/l* [=>] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{1}{\color{blue}{\frac{2}{\frac{D \cdot M}{d}}}}\right)}^{2}}{\ell}} \]
   associate-/r/ [=>] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{1}{2} \cdot \frac{D \cdot M}{d}\right)}}^{2}}{\ell}} \]
   metadata-eval [=>] 85.9	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\color{blue}{0.5} \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}} \]
   associate-/l* [=>] 85.5	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2}}{\ell}} \]
   associate-/r/ [=>] 86.3	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \color{blue}{\left(\frac{D}{d} \cdot M\right)}\right)}^{2}}{\ell}} \]

6. Applied egg-rr 87.4%

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

   [Start] 86.3	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}}{\ell}} \]
   unpow2 [=>] 86.3	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}}{\ell}} \]
   *-un-lft-identity [=>] 86.3	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}{\color{blue}{1 \cdot \ell}}} \]
   times-frac [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{0.5 \cdot \left(\frac{D}{d} \cdot M\right)}{1} \cdot \frac{0.5 \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}\right)}} \]
   *-commutative [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}}{1} \cdot \frac{0.5 \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}\right)} \]
   associate-*l* [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}}{1} \cdot \frac{0.5 \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}\right)} \]
   *-commutative [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}}{\ell}\right)} \]
   associate-*l* [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}}{\ell}\right)} \]

7. Applied egg-rr 88.5%

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

   [Start] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)} \]
   /-rgt-identity [=>] 87.4	\[ w0 \cdot \sqrt{1 - h \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)} \]
   associate-*r* [=>] 88.5	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}}} \]
   associate-*r/ [=>] 88.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}}} \]
   *-commutative [=>] 88.9	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot h\right)} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}} \]
   associate-*r* [=>] 88.9	\[ w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)} \cdot h\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}} \]
   *-commutative [=>] 88.9	\[ w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)} \cdot h\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}} \]
   associate-*l/ [=>] 88.1	\[ w0 \cdot \sqrt{1 - \frac{\left(\left(0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}\right) \cdot h\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}} \]
   associate-*r* [=>] 88.1	\[ w0 \cdot \sqrt{1 - \frac{\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot h\right) \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}}{\ell}} \]
   *-commutative [=>] 88.1	\[ w0 \cdot \sqrt{1 - \frac{\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot h\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}}{\ell}} \]
   associate-*l/ [=>] 88.5	\[ w0 \cdot \sqrt{1 - \frac{\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot h\right) \cdot \left(0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}\right)}{\ell}} \]

8. Final simplification 88.5%

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

Alternatives

Alternative 1
Accuracy	75.1%
Cost	8268

\[\begin{array}{l} t_0 := -0.25 \cdot \left(D \cdot D\right)\\ \mathbf{if}\;D \leq 2 \cdot 10^{-303}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \left(M \cdot \left(\frac{t_0}{d} \cdot \frac{\frac{h}{d}}{\ell}\right)\right)}\\ \mathbf{elif}\;D \leq 3 \cdot 10^{-143}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-0.25}{\frac{d}{\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot h\right)\right)} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;D \leq 2.3 \cdot 10^{+126}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{t_0}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M \cdot h}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot \frac{-1}{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	78.6%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+248}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(M \cdot h\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\ell \cdot \frac{d}{D}}\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-194}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	75.1%
Cost	8140

\[\begin{array}{l} t_0 := -0.25 \cdot \left(D \cdot D\right)\\ \mathbf{if}\;D \leq 2 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \left(M \cdot \left(\frac{t_0}{d} \cdot \frac{\frac{h}{d}}{\ell}\right)\right)}\\ \mathbf{elif}\;D \leq 10^{-144}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-0.25}{\frac{d}{\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot h\right)\right)} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;D \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{t_0}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M \cdot h}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot -0.25\right) \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{h}{d}}{d \cdot \ell}\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	77.1%
Cost	8008

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;M \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{M \cdot t_0}{\frac{d}{D} \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;M \leq -1.3 \cdot 10^{-181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(h \cdot \frac{M}{\ell}\right) \cdot t_0}{\frac{d}{D}}\right)\\ \end{array} \]

Alternative 5
Accuracy	77.9%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;M \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \left(M \cdot \left(\frac{-0.25 \cdot \left(D \cdot D\right)}{d} \cdot \frac{\frac{h}{d}}{\ell}\right)\right)}\\ \mathbf{elif}\;M \leq -1.3 \cdot 10^{-181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(h \cdot \frac{M}{\ell}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{d}{D}}\right)\\ \end{array} \]

Alternative 6
Accuracy	74.9%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;D \leq 4.8 \cdot 10^{-301}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \left(M \cdot \left(\frac{-0.25 \cdot \left(D \cdot D\right)}{d} \cdot \frac{\frac{h}{d}}{\ell}\right)\right)}\\ \mathbf{elif}\;D \leq 1.35 \cdot 10^{+14}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-0.25}{\frac{d}{\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot h\right)\right)} \cdot \frac{d}{M}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot -0.25\right) \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{\frac{h}{d}}{d \cdot \ell}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	72.1%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;D \leq -3 \cdot 10^{-61} \lor \neg \left(D \leq 1.4 \cdot 10^{+28}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(D \cdot \frac{D}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8
Accuracy	73.4%
Cost	1608

\[\begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 400000000000:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9
Accuracy	73.7%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;d \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(h \cdot \frac{M}{\ell}\right) \cdot \frac{M}{\frac{d}{D} \cdot \frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10
Accuracy	76.1%
Cost	1476

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;D \leq -1 \cdot 10^{-303}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{M \cdot t_0}{\frac{d}{D} \cdot \frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(M \cdot h\right) \cdot t_0}{\ell \cdot \frac{d}{D}}\right)\\ \end{array} \]

Alternative 11
Accuracy	75.0%
Cost	1344

\[w0 \cdot \left(1 + -0.125 \cdot \frac{M \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{d}{D} \cdot \frac{\ell}{h}}\right) \]

Alternative 12
Accuracy	67.8%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023163 
(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))))))