Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 88.9%

Time: 19.7s

Alternatives: 4

Speedup: 1.8×

Specification

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Initial Program: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

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

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))));
}

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))))

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

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

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]

Alternative 1: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M \cdot \frac{0.5}{d}\right)\\ w0 \cdot \sqrt{1 - h \cdot \frac{t\_0}{\frac{\ell}{t\_0}}} \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* M (/ 0.5 d)))))
   (* w0 (sqrt (- 1.0 (* h (/ t_0 (/ l t_0))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	return w0 * sqrt((1.0 - (h * (t_0 / (l / t_0)))));
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d_m * (m * (0.5d0 / d))
    code = w0 * sqrt((1.0d0 - (h * (t_0 / (l / t_0)))))
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	return w0 * Math.sqrt((1.0 - (h * (t_0 / (l / t_0)))));
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = D_m * (M * (0.5 / d))
	return w0 * math.sqrt((1.0 - (h * (t_0 / (l / t_0)))))

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M * Float64(0.5 / d)))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 / Float64(l / t_0))))))
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	t_0 = D_m * (M * (0.5 / d));
	tmp = w0 * sqrt((1.0 - (h * (t_0 / (l / t_0)))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 85.1%

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

3. Simplified 83.7%

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

   1. associate-/l* 85.1%

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

   2. clear-num 85.1%

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

   3. un-div-inv 85.1%

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

   4. associate-/l* 83.7%

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

   5. div-inv 83.6%

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

   6. clear-num 83.6%

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

   7. associate-/r* 83.6%

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

6. Applied egg-rr 83.6%

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

   1. associate-/r/ 86.9%

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

   2. *-commutative 86.9%

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

   3. associate-*r/ 88.4%

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

   4. associate-*l/ 88.4%

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

   5. associate-*r/ 88.4%

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

   6. associate-*l/ 88.4%

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

   7. metadata-eval 88.4%

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

   8. associate-/l* 88.4%

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

   9. *-commutative 88.4%

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

   10. associate-*r/ 88.4%

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

   11. associate-*l/ 88.4%

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

   12. associate-*r* 88.4%

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

   13. *-commutative 88.4%

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

   14. *-commutative 88.4%

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

   15. /-rgt-identity 88.4%

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

   16. associate-*r/ 88.4%

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

   17. associate-/l* 88.4%

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

8. Simplified 88.4%

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

   1. div-inv 88.4%

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

   2. associate-*r/ 88.4%

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

   3. div-inv 88.4%

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

   4. clear-num 88.4%

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

   5. associate-*r* 88.4%

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

10. Applied egg-rr 88.4%

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

    1. un-div-inv 88.4%

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

    2. unpow2 88.4%

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

    3. associate-*r/ 90.2%

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

    4. /-rgt-identity 90.2%

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

    5. clear-num 90.2%

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

    6. frac-times 90.2%

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

    7. metadata-eval 90.2%

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

    8. div-inv 90.2%

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

    9. /-rgt-identity 90.2%

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

    10. metadata-eval 90.2%

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

    11. times-frac 90.2%

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

    12. *-un-lft-identity 90.2%

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

    13. *-un-lft-identity 90.2%

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

12. Applied egg-rr 90.2%

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

13. Final simplification 90.2%

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

Alternative 2: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.08 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(w0 \cdot h\right) \cdot \frac{{\left(\frac{D\_m \cdot M}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= D_m 1.08e+45)
   w0
   (+ w0 (* -0.125 (* (* w0 h) (/ (pow (/ (* D_m M) d) 2.0) l))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 1.08e+45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((w0 * h) * (pow(((D_m * M) / d), 2.0) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 1.08d+45) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((w0 * h) * ((((d_m * m) / d) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 1.08e+45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((w0 * h) * (Math.pow(((D_m * M) / d), 2.0) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if D_m <= 1.08e+45:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((w0 * h) * (math.pow(((D_m * M) / d), 2.0) / l)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 1.08e+45)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(w0 * h) * Float64((Float64(Float64(D_m * M) / d) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 1.08e+45)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((w0 * h) * ((((D_m * M) / d) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 1.08e+45], w0, N[(w0 + N[(-0.125 * N[(N[(w0 * h), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.08e45

   1. Initial program 86.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in M around 0 72.5%

      \[\leadsto \color{blue}{w0} \]

   if 1.08e45 < D

   1. Initial program 80.1%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in M around 0 53.0%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. times-frac 56.0%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} \]
   8. Step-by-step derivation

      1. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

      2. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

      3. times-frac 70.6%

         \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

   9. Applied egg-rr 70.6%

      \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

   10. Taylor expanded in D around 0 53.0%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   11. Step-by-step derivation

       1. *-commutative 53.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 53.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 56.0%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \]

       4. associate-/l* 56.0%

          \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \]

       5. associate-*l/ 54.4%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}}{\frac{\ell}{h \cdot w0}}} \]

       6. unpow2 54.4%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}}{\frac{\ell}{h \cdot w0}} \]

       7. unpow2 54.4%

          \[\leadsto w0 + -0.125 \cdot \frac{\left(M \cdot M\right) \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}}{\frac{\ell}{h \cdot w0}} \]

       8. associate-/l/ 58.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}}{\frac{\ell}{h \cdot w0}} \]

       9. unpow2 58.0%

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

       10. associate-*l/ 64.2%

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

       11. associate-*r/ 69.0%

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

       12. swap-sqr 71.4%

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

       13. unpow2 71.4%

           \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}}{\frac{\ell}{h \cdot w0}} \]

       14. associate-/r/ 71.6%

           \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot w0\right)\right)} \]

       15. *-commutative 71.6%

           \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \]

   12. Simplified 71.6%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 73.2%

          \[\leadsto w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right) \]

   14. Applied egg-rr 73.2%

       \[\leadsto w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.08 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(w0 \cdot h\right) \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;D\_m \leq 1.04 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(w0 \cdot h\right) \cdot \frac{t\_0 \cdot t\_0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D_m d))))
   (if (<= D_m 1.04e+45) w0 (+ w0 (* -0.125 (* (* w0 h) (/ (* t_0 t_0) l)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = M * (D_m / d);
	double tmp;
	if (D_m <= 1.04e+45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((w0 * h) * ((t_0 * t_0) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d_m / d)
    if (d_m <= 1.04d+45) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((w0 * h) * ((t_0 * t_0) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = M * (D_m / d);
	double tmp;
	if (D_m <= 1.04e+45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((w0 * h) * ((t_0 * t_0) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = M * (D_m / d)
	tmp = 0
	if D_m <= 1.04e+45:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((w0 * h) * ((t_0 * t_0) / l)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(M * Float64(D_m / d))
	tmp = 0.0
	if (D_m <= 1.04e+45)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(w0 * h) * Float64(Float64(t_0 * t_0) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = M * (D_m / d);
	tmp = 0.0;
	if (D_m <= 1.04e+45)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((w0 * h) * ((t_0 * t_0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 1.04e+45], w0, N[(w0 + N[(-0.125 * N[(N[(w0 * h), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.04e45

   1. Initial program 86.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in M around 0 72.5%

      \[\leadsto \color{blue}{w0} \]

   if 1.04e45 < D

   1. Initial program 80.1%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in M around 0 53.0%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. times-frac 56.0%

         \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} \]
   8. Step-by-step derivation

      1. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

      2. unpow2 56.0%

         \[\leadsto w0 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

      3. times-frac 70.6%

         \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

   9. Applied egg-rr 70.6%

      \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right) \]

   10. Taylor expanded in D around 0 53.0%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   11. Step-by-step derivation

       1. *-commutative 53.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       2. *-commutative 53.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}{\color{blue}{\ell \cdot {d}^{2}}} \]

       3. times-frac 56.0%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \]

       4. associate-/l* 56.0%

          \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \]

       5. associate-*l/ 54.4%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}}{\frac{\ell}{h \cdot w0}}} \]

       6. unpow2 54.4%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}}{\frac{\ell}{h \cdot w0}} \]

       7. unpow2 54.4%

          \[\leadsto w0 + -0.125 \cdot \frac{\left(M \cdot M\right) \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}}{\frac{\ell}{h \cdot w0}} \]

       8. associate-/l/ 58.0%

          \[\leadsto w0 + -0.125 \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}}{\frac{\ell}{h \cdot w0}} \]

       9. unpow2 58.0%

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

       10. associate-*l/ 64.2%

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

       11. associate-*r/ 69.0%

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

       12. swap-sqr 71.4%

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

       13. unpow2 71.4%

           \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}}{\frac{\ell}{h \cdot w0}} \]

       14. associate-/r/ 71.6%

           \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot \left(h \cdot w0\right)\right)} \]

       15. *-commutative 71.6%

           \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \]

   12. Simplified 71.6%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\left(h \cdot w0\right) \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \]
   13. Step-by-step derivation

       1. unpow2 71.6%

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

   14. Applied egg-rr 71.6%

       \[\leadsto w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{\color{blue}{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)}}{\ell}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.04 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(w0 \cdot h\right) \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 216.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d) :precision binary64 w0)

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return w0
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 85.1%

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

3. Simplified 83.7%

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

5. Taylor expanded in M around 0 64.9%

   \[\leadsto \color{blue}{w0} \]

6. Final simplification 64.9%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))