Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 85.8%

Time: 12.6s

Alternatives: 6

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: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}} \end{array} \]

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 (sqrt (- 1.0 (* h (/ (pow (* D_m (* M (/ 0.5 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) {
	return w0 * sqrt((1.0 - (h * (pow((D_m * (M * (0.5 / d))), 2.0) / l))));
}

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 * sqrt((1.0d0 - (h * (((d_m * (m * (0.5d0 / d))) ** 2.0d0) / l))))
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 * Math.sqrt((1.0 - (h * (Math.pow((D_m * (M * (0.5 / d))), 2.0) / l))));
}

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 * math.sqrt((1.0 - (h * (math.pow((D_m * (M * (0.5 / d))), 2.0) / l))))

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 Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)))))
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 * sqrt((1.0 - (h * (((D_m * (M * (0.5 / d))) ^ 2.0) / l))));
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.1%

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

   1. expm1-log1p-u 58.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. expm1-udef 58.2%

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

   3. log1p-udef 58.2%

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

   4. add-exp-log 80.1%

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

   5. +-commutative 80.1%

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

   6. frac-times 80.2%

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

   7. div-inv 80.1%

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

   8. associate-*l* 80.1%

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

   9. associate-/r* 80.1%

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

   10. metadata-eval 80.1%

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

5. Applied egg-rr 80.1%

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

   1. associate--l+ 80.1%

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

   2. metadata-eval 80.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 80.1%

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

   4. associate-*r/ 84.3%

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

   5. associate-*l/ 85.4%

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

   6. *-commutative 85.4%

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

   7. associate-*r* 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-*r* 85.0%

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

7. Simplified 85.0%

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

8. Final simplification 85.0%

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

Alternative 2: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D_m \cdot M}{d}\\ w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right), 1\right) \end{array} \end{array} \]

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 (/ (* D_m M) d)))
   (* w0 (fma -0.125 (* h (* t_0 (/ t_0 l))) 1.0))))

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) / d;
	return w0 * fma(-0.125, (h * (t_0 * (t_0 / l))), 1.0);
}

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(Float64(D_m * M) / d)
	return Float64(w0 * fma(-0.125, Float64(h * Float64(t_0 * Float64(t_0 / l))), 1.0))
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[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision]}, N[(w0 * N[(-0.125 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.1%

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

   1. expm1-log1p-u 58.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. expm1-udef 58.2%

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

   3. log1p-udef 58.2%

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

   4. add-exp-log 80.1%

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

   5. +-commutative 80.1%

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

   6. frac-times 80.2%

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

   7. div-inv 80.1%

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

   8. associate-*l* 80.1%

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

   9. associate-/r* 80.1%

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

   10. metadata-eval 80.1%

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

5. Applied egg-rr 80.1%

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

   1. associate--l+ 80.1%

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

   2. metadata-eval 80.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 80.1%

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

   4. associate-*r/ 84.3%

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

   5. associate-*l/ 85.4%

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

   6. *-commutative 85.4%

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

   7. associate-*r* 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-*r* 85.0%

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

7. Simplified 85.0%

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

8. Taylor expanded in h around 0 56.4%

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

   1. *-commutative 56.4%

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

   2. times-frac 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   5. times-frac 68.2%

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

   6. unpow2 68.2%

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

   7. associate-*l/ 67.4%

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

   8. *-commutative 67.4%

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

10. Simplified 67.4%

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

11. Taylor expanded in D around 0 55.3%

    \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
12. Step-by-step derivation

    1. +-commutative 55.3%

       \[\leadsto w0 \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

    2. fma-def 55.3%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]

13. Simplified 78.6%

    \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right)} \]
14. Step-by-step derivation

    1. unpow2 78.6%

       \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{\ell}, 1\right) \]

    2. *-un-lft-identity 78.6%

       \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{1 \cdot \ell}}, 1\right) \]

    3. times-frac 80.2%

       \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{d}}{1} \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)}, 1\right) \]

    4. associate-*r/ 79.1%

       \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(\frac{\color{blue}{\frac{D \cdot M}{d}}}{1} \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right), 1\right) \]

    5. associate-*r/ 79.8%

       \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(\frac{\frac{D \cdot M}{d}}{1} \cdot \frac{\color{blue}{\frac{D \cdot M}{d}}}{\ell}\right), 1\right) \]

15. Applied egg-rr 79.8%

    \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \color{blue}{\left(\frac{\frac{D \cdot M}{d}}{1} \cdot \frac{\frac{D \cdot M}{d}}{\ell}\right)}, 1\right) \]

16. Final simplification 79.8%

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

Alternative 3: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 29500:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D_m \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= M 29500.0)
   w0
   (* w0 (* (/ h l) (* -0.125 (pow (* D_m (/ M d)) 2.0))))))

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 tmp;
	if (M <= 29500.0) {
		tmp = w0;
	} else {
		tmp = w0 * ((h / l) * (-0.125 * pow((D_m * (M / d)), 2.0)));
	}
	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 (m <= 29500.0d0) then
        tmp = w0
    else
        tmp = w0 * ((h / l) * ((-0.125d0) * ((d_m * (m / d)) ** 2.0d0)))
    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 (M <= 29500.0) {
		tmp = w0;
	} else {
		tmp = w0 * ((h / l) * (-0.125 * Math.pow((D_m * (M / d)), 2.0)));
	}
	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 M <= 29500.0:
		tmp = w0
	else:
		tmp = w0 * ((h / l) * (-0.125 * math.pow((D_m * (M / d)), 2.0)))
	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 (M <= 29500.0)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(h / l) * Float64(-0.125 * (Float64(D_m * Float64(M / d)) ^ 2.0))));
	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 (M <= 29500.0)
		tmp = w0;
	else
		tmp = w0 * ((h / l) * (-0.125 * ((D_m * (M / d)) ^ 2.0)));
	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[M, 29500.0], w0, N[(w0 * N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 29500

   1. Initial program 82.1%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 29500 < M

   1. Initial program 73.7%

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

   3. Simplified 75.3%

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

      1. expm1-log1p-u 37.3%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-udef 37.3%

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

      3. log1p-udef 37.3%

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

      4. add-exp-log 75.3%

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

      5. +-commutative 75.3%

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

      6. frac-times 73.7%

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

      7. div-inv 73.7%

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

      8. associate-*l* 75.2%

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

      9. associate-/r* 75.2%

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

      10. metadata-eval 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. associate--l+ 75.2%

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

      2. metadata-eval 75.2%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 75.2%

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

      4. associate-*r/ 75.6%

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

      5. associate-*l/ 77.1%

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

      6. *-commutative 77.1%

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

      7. associate-*r* 75.6%

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

      8. *-commutative 75.6%

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

      9. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in h around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. times-frac 39.9%

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

      3. unpow2 39.9%

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

      4. unpow2 39.9%

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

      5. times-frac 50.5%

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

      6. unpow2 50.5%

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

      7. associate-*l/ 47.3%

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

      8. *-commutative 47.3%

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

   10. Simplified 47.3%

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

   11. Taylor expanded in D around 0 41.5%

       \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   12. Step-by-step derivation

       1. +-commutative 41.5%

          \[\leadsto w0 \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

       2. fma-def 41.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]

   13. Simplified 63.0%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right)} \]

   14. Taylor expanded in h around inf 23.4%

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

   16. Simplified 28.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 29500:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 + w0 \cdot \left(-0.125 \cdot \frac{h \cdot {\left(\frac{D_m \cdot M}{d}\right)}^{2}}{\ell}\right) \end{array} \]

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 (* w0 (* -0.125 (/ (* 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) {
	return w0 + (w0 * (-0.125 * ((h * pow(((D_m * M) / d), 2.0)) / l)));
}

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 + (w0 * ((-0.125d0) * ((h * (((d_m * m) / d) ** 2.0d0)) / l)))
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 + (w0 * (-0.125 * ((h * Math.pow(((D_m * M) / d), 2.0)) / l)));
}

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 + (w0 * (-0.125 * ((h * math.pow(((D_m * M) / d), 2.0)) / l)))

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 Float64(w0 + Float64(w0 * Float64(-0.125 * Float64(Float64(h * (Float64(Float64(D_m * M) / d) ^ 2.0)) / l))))
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 + (w0 * (-0.125 * ((h * (((D_m * M) / d) ^ 2.0)) / l)));
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_] := N[(w0 + N[(w0 * N[(-0.125 * N[(N[(h * N[Power[N[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.1%

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

   1. expm1-log1p-u 58.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. expm1-udef 58.2%

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

   3. log1p-udef 58.2%

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

   4. add-exp-log 80.1%

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

   5. +-commutative 80.1%

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

   6. frac-times 80.2%

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

   7. div-inv 80.1%

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

   8. associate-*l* 80.1%

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

   9. associate-/r* 80.1%

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

   10. metadata-eval 80.1%

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

5. Applied egg-rr 80.1%

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

   1. associate--l+ 80.1%

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

   2. metadata-eval 80.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 80.1%

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

   4. associate-*r/ 84.3%

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

   5. associate-*l/ 85.4%

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

   6. *-commutative 85.4%

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

   7. associate-*r* 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-*r* 85.0%

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

7. Simplified 85.0%

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

8. Taylor expanded in h around 0 56.4%

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

   1. *-commutative 56.4%

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

   2. times-frac 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   5. times-frac 68.2%

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

   6. unpow2 68.2%

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

   7. associate-*l/ 67.4%

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

   8. *-commutative 67.4%

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

10. Simplified 67.4%

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

11. Taylor expanded in D around 0 55.3%

    \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
12. Step-by-step derivation

    1. +-commutative 55.3%

       \[\leadsto w0 \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

    2. fma-def 55.3%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]

13. Simplified 78.6%

    \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right)} \]
14. Step-by-step derivation

    1. add-sqr-sqrt 43.1%

       \[\leadsto \color{blue}{\left(\sqrt{w0} \cdot \sqrt{w0}\right)} \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right) \]

    2. unpow2 43.1%

       \[\leadsto \color{blue}{{\left(\sqrt{w0}\right)}^{2}} \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right) \]

    3. fma-udef 43.1%

       \[\leadsto {\left(\sqrt{w0}\right)}^{2} \cdot \color{blue}{\left(-0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) + 1\right)} \]

    4. distribute-rgt-in 43.1%

       \[\leadsto \color{blue}{\left(-0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right) \cdot {\left(\sqrt{w0}\right)}^{2} + 1 \cdot {\left(\sqrt{w0}\right)}^{2}} \]

    5. associate-*r/ 43.1%

       \[\leadsto \left(-0.125 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}\right) \cdot {\left(\sqrt{w0}\right)}^{2} + 1 \cdot {\left(\sqrt{w0}\right)}^{2} \]

    6. associate-*r/ 43.1%

       \[\leadsto \left(-0.125 \cdot \frac{h \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right) \cdot {\left(\sqrt{w0}\right)}^{2} + 1 \cdot {\left(\sqrt{w0}\right)}^{2} \]

    7. unpow2 43.1%

       \[\leadsto \left(-0.125 \cdot \frac{h \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot \color{blue}{\left(\sqrt{w0} \cdot \sqrt{w0}\right)} + 1 \cdot {\left(\sqrt{w0}\right)}^{2} \]

    8. add-sqr-sqrt 43.1%

       \[\leadsto \left(-0.125 \cdot \frac{h \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot \color{blue}{w0} + 1 \cdot {\left(\sqrt{w0}\right)}^{2} \]

    9. unpow2 43.1%

       \[\leadsto \left(-0.125 \cdot \frac{h \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right) \cdot w0 + 1 \cdot \color{blue}{\left(\sqrt{w0} \cdot \sqrt{w0}\right)} \]

    10. *-un-lft-identity 43.1%

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

    11. add-sqr-sqrt 78.9%

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

15. Applied egg-rr 78.9%

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

16. Final simplification 78.9%

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

Alternative 5: 67.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{w0 \cdot w0}\\ \end{array} \end{array} \]

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 (<= M 7.1e-6) w0 (sqrt (* w0 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) {
	double tmp;
	if (M <= 7.1e-6) {
		tmp = w0;
	} else {
		tmp = sqrt((w0 * w0));
	}
	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 (m <= 7.1d-6) then
        tmp = w0
    else
        tmp = sqrt((w0 * w0))
    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 (M <= 7.1e-6) {
		tmp = w0;
	} else {
		tmp = Math.sqrt((w0 * w0));
	}
	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 M <= 7.1e-6:
		tmp = w0
	else:
		tmp = math.sqrt((w0 * w0))
	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 (M <= 7.1e-6)
		tmp = w0;
	else
		tmp = sqrt(Float64(w0 * w0));
	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 (M <= 7.1e-6)
		tmp = w0;
	else
		tmp = sqrt((w0 * w0));
	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[M, 7.1e-6], w0, N[Sqrt[N[(w0 * w0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.0999999999999998e-6

   1. Initial program 81.8%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 73.4%

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

   if 7.0999999999999998e-6 < M

   1. Initial program 75.0%

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

   3. Simplified 76.5%

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

      1. add-sqr-sqrt 37.3%

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

      2. sqrt-unprod 28.9%

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

      3. *-commutative 28.9%

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

      4. *-commutative 28.9%

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

      5. swap-sqr 25.4%

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

   5. Applied egg-rr 25.4%

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

      1. *-commutative 25.4%

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

      2. associate-*r/ 25.4%

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

      3. *-commutative 25.4%

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

      4. associate-*r/ 25.4%

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

      5. associate-/l* 25.4%

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

      6. *-commutative 25.4%

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

   7. Simplified 25.4%

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

   8. Taylor expanded in h around 0 15.4%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 15.4%

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

   10. Applied egg-rr 15.4%

       \[\leadsto \sqrt{\color{blue}{w0 \cdot w0}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.1 \cdot 10^{-6}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{w0 \cdot w0}\\ \end{array} \]

Alternative 6: 68.1% 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 80.2%

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

3. Simplified 80.1%

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

4. Taylor expanded in M around 0 66.9%

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

5. Final simplification 66.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023330 
(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))))))