Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 94.0%

Time: 16.3s

Alternatives: 5

Speedup: 0.9×

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: 81.2% 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: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+64}:\\ \;\;\;\;w0 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{4}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M d))))
   (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+64)
     (* w0 (* D (* (/ M d) (sqrt (/ (* h -0.25) l)))))
     (* w0 (sqrt (- 1.0 (* h (* (/ t_0 l) (/ t_0 4.0)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64) {
		tmp = w0 * (D * ((M / d) * sqrt(((h * -0.25) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * ((t_0 / l) * (t_0 / 4.0)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (m / d_1)
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+64)) then
        tmp = w0 * (d * ((m / d_1) * sqrt(((h * (-0.25d0)) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * ((t_0 / l) * (t_0 / 4.0d0)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64) {
		tmp = w0 * (D * ((M / d) * Math.sqrt(((h * -0.25) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * ((t_0 / l) * (t_0 / 4.0)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = D * (M / d)
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+64:
		tmp = w0 * (D * ((M / d) * math.sqrt(((h * -0.25) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * ((t_0 / l) * (t_0 / 4.0)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+64)
		tmp = Float64(w0 * Float64(D * Float64(Float64(M / d) * sqrt(Float64(Float64(h * -0.25) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / 4.0))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+64)
		tmp = w0 * (D * ((M / d) * sqrt(((h * -0.25) / l))));
	else
		tmp = w0 * sqrt((1.0 - (h * ((t_0 / l) * (t_0 / 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+64], N[(w0 * N[(D * N[(N[(M / d), $MachinePrecision] * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -5e64

   1. Initial program 70.0%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in D around inf 41.7%

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

      1. *-commutative 41.7%

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

      2. times-frac 41.7%

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

   6. Simplified 41.7%

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

      1. expm1-log1p-u 41.4%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.25}\right)\right)} \]

      2. expm1-udef 41.4%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.25}\right)} - 1\right)} \]

      3. frac-times 41.6%

         \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot -0.25}\right)} - 1\right) \]

      4. *-commutative 41.6%

         \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot -0.25}\right)} - 1\right) \]

   8. Applied egg-rr 41.6%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell} \cdot -0.25}\right)} - 1\right)} \]

   10. Simplified 68.9%

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

       1. expm1-log1p-u 67.7%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)\right)} \]

       2. expm1-udef 67.7%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot \left(-0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right)} - 1\right)} \]

       3. associate-*r* 67.7%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{h}{\ell} \cdot -0.25\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}\right)} - 1\right) \]

       4. sqrt-prod 71.7%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right)} - 1\right) \]

       5. unpow2 71.7%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \sqrt{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}\right)} - 1\right) \]

       6. sqrt-prod 39.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \color{blue}{\left(\sqrt{D \cdot \frac{M}{d}} \cdot \sqrt{D \cdot \frac{M}{d}}\right)}\right)} - 1\right) \]

       7. add-sqr-sqrt 39.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)} - 1\right) \]

   12. Applied egg-rr 39.4%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \left(D \cdot \frac{M}{d}\right)\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 39.4%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)} \]

       2. expm1-log1p 40.9%

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

       3. *-commutative 40.9%

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

       4. associate-*l* 38.7%

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

       5. *-commutative 38.7%

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

       6. associate-*r/ 38.7%

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

   14. Simplified 38.7%

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

   if -5e64 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 86.2%

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

   3. Simplified 86.2%

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

      1. associate-*r/ 96.8%

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

      2. frac-times 96.8%

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

      3. *-commutative 96.8%

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

      4. unpow2 96.8%

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

      5. unpow2 96.8%

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

      6. *-un-lft-identity 96.8%

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

      7. times-frac 96.8%

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

      8. metadata-eval 96.8%

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

      9. *-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in D around 0 58.1%

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

      1. associate-/l* 58.2%

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

      2. *-commutative 58.2%

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

      3. associate-/l* 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-*r* 61.2%

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

      6. unpow2 61.2%

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

      7. unpow2 61.2%

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

      8. swap-sqr 76.6%

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

      9. unpow2 76.6%

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

      10. times-frac 69.0%

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

      11. unpow2 69.0%

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

      12. unpow2 69.0%

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

      13. times-frac 86.2%

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

      14. associate-*l/ 85.3%

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

      15. associate-*l/ 85.3%

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

      16. unpow2 85.3%

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

   8. Simplified 96.8%

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

      1. unpow2 96.8%

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

      2. div-inv 96.8%

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

      3. times-frac 98.3%

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

      4. metadata-eval 98.3%

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

   10. Applied egg-rr 98.3%

       \[\leadsto w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{d}}{\ell} \cdot \frac{D \cdot \frac{M}{d}}{4}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+64}:\\ \;\;\;\;w0 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{D \cdot \frac{M}{d}}{\ell} \cdot \frac{D \cdot \frac{M}{d}}{4}\right)}\\ \end{array} \]

Alternative 2: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 7.7 \cdot 10^{+151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 7.7e+151)
   w0
   (+ w0 (* -0.125 (* (pow (* D (/ M d)) 2.0) (/ (* h w0) l))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 7.7e+151) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D * (M / d)), 2.0) * ((h * w0) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 7.7d+151) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * (m / d_1)) ** 2.0d0) * ((h * w0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 7.7e+151) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D * (M / d)), 2.0) * ((h * w0) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 7.7e+151:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D * (M / d)), 2.0) * ((h * w0) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 7.7e+151)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(Float64(h * w0) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 7.7e+151)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * (M / d)) ^ 2.0) * ((h * w0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 7.7e+151], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 7.70000000000000038e151

   1. Initial program 83.3%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in D around 0 68.9%

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

   if 7.70000000000000038e151 < D

   1. Initial program 62.9%

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

   3. Simplified 63.0%

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

      1. associate-*r/ 77.1%

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

      2. frac-times 77.0%

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

      3. *-commutative 77.0%

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

      4. unpow2 77.0%

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

      5. unpow2 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. times-frac 77.0%

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

      8. metadata-eval 77.0%

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

      9. *-commutative 77.0%

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

   5. Applied egg-rr 77.0%

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

   6. Taylor expanded in D around 0 39.0%

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

      1. associate-*r* 39.1%

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

      2. unpow2 39.1%

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

      3. unpow2 39.1%

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

      4. swap-sqr 56.7%

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

      5. unpow2 56.7%

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

   8. Simplified 56.7%

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

      1. distribute-rgt-in 56.7%

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

      2. *-un-lft-identity 56.7%

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

      3. *-commutative 56.7%

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

      4. times-frac 39.5%

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

      5. unpow2 39.5%

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

      6. unpow2 39.5%

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

      7. frac-times 53.5%

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

      8. associate-*r/ 53.5%

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

      9. associate-*r/ 53.5%

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

      10. unpow2 53.5%

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

   10. Applied egg-rr 53.5%

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

   11. Taylor expanded in D around 0 39.1%

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

       1. associate-*r* 39.1%

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

       2. times-frac 35.4%

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

   13. Simplified 63.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.7 \cdot 10^{+151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.9× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (+ w0 (* w0 (* (/ (* h (pow (/ M (/ d D)) 2.0)) l) -0.125))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (w0 * (((h * pow((M / (d / D)), 2.0)) / l) * -0.125));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 + (w0 * (((h * ((m / (d_1 / d)) ** 2.0d0)) / l) * (-0.125d0)))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (w0 * (((h * Math.pow((M / (d / D)), 2.0)) / l) * -0.125));
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 + (w0 * (((h * math.pow((M / (d / D)), 2.0)) / l) * -0.125))

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 + Float64(w0 * Float64(Float64(Float64(h * (Float64(M / Float64(d / D)) ^ 2.0)) / l) * -0.125)))
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 + (w0 * (((h * ((M / (d / D)) ^ 2.0)) / l) * -0.125));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 + N[(w0 * N[(N[(N[(h * N[Power[N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $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 80.6%

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

   1. associate-*r/ 88.2%

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

   2. frac-times 88.6%

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

   3. *-commutative 88.6%

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

   4. unpow2 88.6%

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

   5. unpow2 88.6%

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

   6. *-un-lft-identity 88.6%

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

   7. times-frac 88.6%

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

   8. metadata-eval 88.6%

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

   9. *-commutative 88.6%

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

5. Applied egg-rr 88.6%

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

6. Taylor expanded in D around 0 51.7%

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

   1. associate-*r* 54.6%

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

   2. unpow2 54.6%

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

   3. unpow2 54.6%

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

   4. swap-sqr 66.0%

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

   5. unpow2 66.0%

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

8. Simplified 66.0%

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

   1. distribute-rgt-in 66.0%

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

   2. *-un-lft-identity 66.0%

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

   3. *-commutative 66.0%

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

   4. times-frac 61.3%

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

   5. unpow2 61.3%

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

   6. unpow2 61.3%

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

   7. frac-times 74.4%

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

   8. associate-*r/ 74.4%

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

   9. associate-*r/ 74.4%

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

   10. unpow2 74.4%

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

10. Applied egg-rr 74.4%

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

    1. associate-*r/ 81.6%

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

    2. *-commutative 81.6%

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

    3. associate-/r/ 81.5%

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

12. Applied egg-rr 81.5%

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

13. Final simplification 81.5%

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

Alternative 4: 78.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{\frac{d}{D}}\\ \mathbf{if}\;D \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ M (/ d D))))
   (if (<= D 3.2e+25) w0 (+ w0 (* w0 (* -0.125 (* (/ h l) (* t_0 t_0))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d / D);
	double tmp;
	if (D <= 3.2e+25) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m / (d_1 / d)
    if (d <= 3.2d+25) then
        tmp = w0
    else
        tmp = w0 + (w0 * ((-0.125d0) * ((h / l) * (t_0 * t_0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d / D);
	double tmp;
	if (D <= 3.2e+25) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = M / (d / D)
	tmp = 0
	if D <= 3.2e+25:
		tmp = w0
	else:
		tmp = w0 + (w0 * (-0.125 * ((h / l) * (t_0 * t_0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(M / Float64(d / D))
	tmp = 0.0
	if (D <= 3.2e+25)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(-0.125 * Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M / (d / D);
	tmp = 0.0;
	if (D <= 3.2e+25)
		tmp = w0;
	else
		tmp = w0 + (w0 * (-0.125 * ((h / l) * (t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 3.2e+25], w0, N[(w0 + N[(w0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 3.1999999999999999e25

   1. Initial program 82.7%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in D around 0 70.5%

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

   if 3.1999999999999999e25 < D

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 82.2%

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

      2. frac-times 82.2%

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

      3. *-commutative 82.2%

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

      4. unpow2 82.2%

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

      5. unpow2 82.2%

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

      6. *-un-lft-identity 82.2%

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

      7. times-frac 82.2%

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

      8. metadata-eval 82.2%

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

      9. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   6. Taylor expanded in D around 0 51.8%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

      3. unpow2 52.4%

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

      4. swap-sqr 62.0%

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

      5. unpow2 62.0%

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

   8. Simplified 62.0%

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

      1. distribute-rgt-in 62.0%

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

      2. *-un-lft-identity 62.0%

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

      3. *-commutative 62.0%

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

      4. times-frac 52.6%

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

      5. unpow2 52.6%

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

      6. unpow2 52.6%

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

      7. frac-times 65.9%

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

      8. associate-*r/ 65.9%

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

      9. associate-*r/ 65.9%

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

      10. unpow2 65.9%

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

   10. Applied egg-rr 65.9%

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

       1. unpow2 65.9%

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

       2. *-commutative 65.9%

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

       3. associate-/r/ 65.9%

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

       4. *-commutative 65.9%

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

       5. associate-/r/ 65.9%

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

   12. Applied egg-rr 65.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)\right)\right)\\ \end{array} \]

Alternative 5: 68.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, 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 80.6%

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

4. Taylor expanded in D around 0 65.0%

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

5. Final simplification 65.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023299 
(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))))))