Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 89.8%

Time: 18.1s

Alternatives: 5

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: 80.8% 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: 89.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Simplified 81.4%

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

   1. unpow2 81.4%

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

   2. unpow2 81.4%

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

   3. associate-/r* 81.4%

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

   4. associate-*r/ 81.0%

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

   5. *-commutative 81.0%

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

   6. associate-*r/ 85.5%

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

   7. *-commutative 85.5%

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

   8. associate-*l/ 85.5%

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

   9. associate-/r* 85.5%

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

   10. clear-num 85.5%

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

   11. associate-*l/ 85.8%

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

   12. *-un-lft-identity 85.8%

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

   13. associate-/r/ 85.7%

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

6. Applied egg-rr 85.7%

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

   1. associate-*l/ 81.6%

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

   2. unpow2 81.6%

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

   3. associate-*r* 83.4%

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

   4. div-inv 83.2%

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

   5. associate-*l/ 83.2%

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

   6. *-commutative 83.2%

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

   7. clear-num 83.2%

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

   8. *-un-lft-identity 83.2%

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

   9. times-frac 83.2%

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

   10. metadata-eval 83.2%

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

   11. div-inv 83.2%

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

   12. associate-*l/ 83.2%

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

   13. *-commutative 83.2%

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

   14. clear-num 83.2%

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

   15. *-un-lft-identity 83.2%

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

   16. times-frac 83.2%

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

   17. metadata-eval 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 88.1%

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

   2. associate-*r/ 88.1%

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

10. Applied egg-rr 88.1%

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

    1. associate-/l* 88.5%

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

    2. associate-/l* 88.5%

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

12. Simplified 88.5%

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

13. Final simplification 88.5%

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

Alternative 2: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \left(0.5 \cdot \frac{D\_m \cdot \left(h \cdot M\_m\right)}{d \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 2.4e-28)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* D_m (* 0.5 (/ M_m d))) (* 0.5 (/ (* D_m (* h M_m)) (* d l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2.4e-28) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m * (0.5 * (M_m / d))) * (0.5 * ((D_m * (h * M_m)) / (d * l))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 <= 2.4d-28) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((d_m * (0.5d0 * (m_m / d))) * (0.5d0 * ((d_m * (h * m_m)) / (d * l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2.4e-28) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * (0.5 * (M_m / d))) * (0.5 * ((D_m * (h * M_m)) / (d * l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 2.4e-28:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((D_m * (0.5 * (M_m / d))) * (0.5 * ((D_m * (h * M_m)) / (d * l))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 2.4e-28)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * Float64(0.5 * Float64(Float64(D_m * Float64(h * M_m)) / Float64(d * l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 2.4e-28)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((D_m * (0.5 * (M_m / d))) * (0.5 * ((D_m * (h * M_m)) / (d * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 2.4e-28], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(D$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.4000000000000002e-28

   1. Initial program 82.5%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 75.1%

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

   if 2.4000000000000002e-28 < D

   1. Initial program 75.8%

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

   3. Simplified 79.0%

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

      1. unpow2 79.0%

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

      2. unpow2 79.0%

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

      3. associate-/r* 79.0%

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

      4. associate-*r/ 75.8%

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

      5. *-commutative 75.8%

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

      6. associate-*r/ 77.7%

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

      7. *-commutative 77.7%

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

      8. associate-*l/ 80.9%

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

      9. associate-/r* 80.9%

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

      10. clear-num 80.9%

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

      11. associate-*l/ 81.0%

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

      12. *-un-lft-identity 81.0%

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

      13. associate-/r/ 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-*l/ 79.1%

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

      2. unpow2 79.1%

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

      3. associate-*r* 82.5%

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

      4. div-inv 82.5%

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

      5. associate-*l/ 82.5%

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

      6. *-commutative 82.5%

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

      7. clear-num 82.5%

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

      8. *-un-lft-identity 82.5%

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

      9. times-frac 82.5%

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

      10. metadata-eval 82.5%

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

      11. div-inv 82.4%

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

      12. associate-*l/ 82.4%

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

      13. *-commutative 82.4%

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

      14. clear-num 82.4%

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

      15. *-un-lft-identity 82.4%

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

      16. times-frac 82.4%

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

      17. metadata-eval 82.4%

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

   8. Applied egg-rr 82.4%

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

   9. Taylor expanded in h around 0 79.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \left(0.5 \cdot \frac{D \cdot \left(h \cdot M\right)}{d \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D\_m \cdot \left(h \cdot M\_m\right)}{d \cdot \ell}\right) \cdot \left(0.5 \cdot \frac{D\_m \cdot M\_m}{d}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 6.5e-29)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* 0.5 (/ (* D_m (* h M_m)) (* d l))) (* 0.5 (/ (* D_m M_m) d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 6.5e-29) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((0.5 * ((D_m * (h * M_m)) / (d * l))) * (0.5 * ((D_m * M_m) / d)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 <= 6.5d-29) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * ((d_m * (h * m_m)) / (d * l))) * (0.5d0 * ((d_m * m_m) / d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 6.5e-29) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * ((D_m * (h * M_m)) / (d * l))) * (0.5 * ((D_m * M_m) / d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 6.5e-29:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * ((D_m * (h * M_m)) / (d * l))) * (0.5 * ((D_m * M_m) / d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 6.5e-29)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(Float64(D_m * Float64(h * M_m)) / Float64(d * l))) * Float64(0.5 * Float64(Float64(D_m * M_m) / d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 6.5e-29)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((0.5 * ((D_m * (h * M_m)) / (d * l))) * (0.5 * ((D_m * M_m) / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 6.5e-29], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(N[(D$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.5e-29

   1. Initial program 82.5%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 75.1%

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

   if 6.5e-29 < D

   1. Initial program 75.8%

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

   3. Simplified 79.0%

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

      1. unpow2 79.0%

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

      2. unpow2 79.0%

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

      3. associate-/r* 79.0%

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

      4. associate-*r/ 75.8%

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

      5. *-commutative 75.8%

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

      6. associate-*r/ 77.7%

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

      7. *-commutative 77.7%

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

      8. associate-*l/ 80.9%

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

      9. associate-/r* 80.9%

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

      10. clear-num 80.9%

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

      11. associate-*l/ 81.0%

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

      12. *-un-lft-identity 81.0%

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

      13. associate-/r/ 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-*l/ 79.1%

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

      2. unpow2 79.1%

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

      3. associate-*r* 82.5%

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

      4. div-inv 82.5%

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

      5. associate-*l/ 82.5%

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

      6. *-commutative 82.5%

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

      7. clear-num 82.5%

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

      8. *-un-lft-identity 82.5%

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

      9. times-frac 82.5%

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

      10. metadata-eval 82.5%

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

      11. div-inv 82.4%

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

      12. associate-*l/ 82.4%

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

      13. *-commutative 82.4%

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

      14. clear-num 82.4%

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

      15. *-un-lft-identity 82.4%

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

      16. times-frac 82.4%

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

      17. metadata-eval 82.4%

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

   8. Applied egg-rr 82.4%

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

   9. Taylor expanded in h around 0 79.3%

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

   10. Taylor expanded in D around 0 74.3%

       \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D \cdot \left(h \cdot M\right)}{d \cdot \ell}\right) \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Simplified 81.4%

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

   1. unpow2 81.4%

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

   2. unpow2 81.4%

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

   3. associate-/r* 81.4%

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

   4. associate-*r/ 81.0%

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

   5. *-commutative 81.0%

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

   6. associate-*r/ 85.5%

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

   7. *-commutative 85.5%

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

   8. associate-*l/ 85.5%

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

   9. associate-/r* 85.5%

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

   10. clear-num 85.5%

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

   11. associate-*l/ 85.8%

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

   12. *-un-lft-identity 85.8%

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

   13. associate-/r/ 85.7%

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

6. Applied egg-rr 85.7%

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

   1. associate-*l/ 81.6%

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

   2. unpow2 81.6%

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

   3. associate-*r* 83.4%

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

   4. div-inv 83.2%

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

   5. associate-*l/ 83.2%

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

   6. *-commutative 83.2%

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

   7. clear-num 83.2%

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

   8. *-un-lft-identity 83.2%

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

   9. times-frac 83.2%

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

   10. metadata-eval 83.2%

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

   11. div-inv 83.2%

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

   12. associate-*l/ 83.2%

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

   13. *-commutative 83.2%

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

   14. clear-num 83.2%

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

   15. *-un-lft-identity 83.2%

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

   16. times-frac 83.2%

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

   17. metadata-eval 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 88.1%

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

   2. associate-*r/ 88.1%

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

10. Applied egg-rr 88.1%

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

    1. associate-/l* 88.5%

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

    2. associate-/l* 88.5%

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

12. Simplified 88.5%

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

    1. associate-/l* 86.2%

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

    2. associate-*r/ 86.2%

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

14. Applied egg-rr 86.2%

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

15. Final simplification 86.2%

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

Alternative 5: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

3. Simplified 81.4%

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

5. Taylor expanded in D around 0 68.4%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))))