Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 85.4%

Time: 17.4s

Alternatives: 16

Speedup: 1.0×

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: 80.7% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq -4 \cdot 10^{+195}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M}{\frac{\ell}{M \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) -4e+195)
   (* w0 (sqrt (- 1.0 (* 0.25 (* (pow (/ D d) 2.0) (/ M (/ l (* M h))))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= -4e+195) {
		tmp = w0 * sqrt((1.0 - (0.25 * (pow((D / d), 2.0) * (M / (l / (M * h)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	}
	return tmp;
}

Fortran

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 ((m * d) <= (-4d+195)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) ** 2.0d0) * (m / (l / (m * h)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= -4e+195) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (Math.pow((D / d), 2.0) * (M / (l / (M * h)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M * D) <= -4e+195:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (math.pow((D / d), 2.0) * (M / (l / (M * h)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= -4e+195)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64((Float64(D / d) ^ 2.0) * Float64(M / Float64(l / Float64(M * h))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M * D) <= -4e+195)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) ^ 2.0) * (M / (l / (M * h)))))));
	else
		tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], -4e+195], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M / N[(l / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < -3.99999999999999991e195

   1. Initial program 42.4%

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

      1. *-commutative 42.4%

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

      2. times-frac 42.4%

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

   3. Simplified 42.4%

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

   4. Taylor expanded in M around 0 28.7%

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

      1. associate-*r/ 28.7%

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

      2. *-commutative 28.7%

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

      3. times-frac 29.5%

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

      4. associate-*r* 29.7%

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

      5. unpow2 29.7%

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

      6. associate-*l* 33.8%

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

      7. unpow2 33.8%

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

      8. unpow2 33.8%

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

   6. Simplified 33.8%

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

      1. associate-*r/ 33.8%

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

      2. associate-*l* 33.8%

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

      3. *-commutative 33.8%

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

   8. Applied egg-rr 33.8%

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

      1. *-un-lft-identity 33.8%

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

      2. times-frac 38.2%

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

      3. *-commutative 38.2%

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

   10. Applied egg-rr 38.2%

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

       1. *-lft-identity 38.2%

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

       2. associate-/r* 38.2%

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

       3. associate-/l* 42.3%

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

       4. associate-/r/ 42.3%

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

       5. *-commutative 42.3%

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

       6. *-commutative 42.3%

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

       7. associate-*r* 42.3%

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

   12. Simplified 42.3%

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

   13. Taylor expanded in l around 0 28.7%

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

       1. times-frac 29.5%

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

       2. unpow2 29.5%

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

       3. unpow2 29.5%

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

       4. times-frac 38.3%

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

       5. unpow2 38.3%

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

       6. unpow2 38.3%

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

       7. associate-*r* 38.3%

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

       8. associate-/l* 46.6%

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

       9. *-commutative 46.6%

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

   15. Simplified 46.6%

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

   if -3.99999999999999991e195 < (*.f64 M D)

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. frac-times 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-*l/ 89.3%

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

      5. *-commutative 89.3%

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

      6. frac-times 89.7%

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

      7. div-inv 89.7%

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

      8. metadata-eval 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq -4 \cdot 10^{+195}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M}{\frac{\ell}{M \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.02:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) 0.02)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* D 0.5) (/ d M)) 2.0)))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 0.02) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D * 0.5) / (d / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= 0.02d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * 0.5d0) / (d_1 / m)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 0.02) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D * 0.5) / (d / M)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= 0.02:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D * 0.5) / (d / M)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= 0.02)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D * 0.5) / Float64(d / M)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= 0.02)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * 0.5) / (d / M)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.02], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 0.0200000000000000004

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. times-frac 86.2%

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

   3. Simplified 86.2%

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

      1. *-commutative 86.2%

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

      2. clear-num 86.2%

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

      3. un-div-inv 86.7%

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

      4. div-inv 86.7%

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

      5. metadata-eval 86.7%

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

   5. Applied egg-rr 86.7%

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

   if 0.0200000000000000004 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in M around 0 73.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.02:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-166}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (/ (* (* D (* D h)) (/ 0.25 l)) (* (/ d M) (/ d M))))))
   (if (<= (/ h l) -1e-166)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (* M D) (/ 0.5 d)) 2.0)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -1e-166) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M * D) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -1e-166) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M * D) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))))
	elif (h / l) <= -1e-166:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M * D) * (0.5 / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * Float64(D * h)) * Float64(0.25 / l)) / Float64(Float64(d / M) * Float64(d / M))))));
	elseif (Float64(h / l) <= -1e-166)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M * D) * Float64(0.5 / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	elseif ((h / l) <= -1e-166)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M * D) * (0.5 / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-166], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. times-frac 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in M around 0 50.9%

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

      1. associate-*r/ 50.9%

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

      2. *-commutative 50.9%

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

      3. times-frac 55.6%

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

      4. associate-*r* 51.1%

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

      5. unpow2 51.1%

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

      6. associate-*l* 55.2%

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

      7. unpow2 55.2%

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

      8. unpow2 55.2%

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

   6. Simplified 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. associate-/l* 55.2%

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

      4. *-commutative 55.2%

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

   8. Applied egg-rr 55.2%

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

   10. Simplified 69.6%

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

   if -inf.0 < (/.f64 h l) < -1.00000000000000004e-166

   1. Initial program 81.1%

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

      1. associate-/l* 82.0%

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

   3. Simplified 82.0%

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

      1. associate-/l* 81.1%

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

      2. clear-num 81.0%

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

      3. associate-/r/ 81.1%

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

      4. associate-/r* 81.1%

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

      5. metadata-eval 81.1%

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

   5. Applied egg-rr 81.1%

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

   if -1.00000000000000004e-166 < (/.f64 h l)

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. times-frac 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 90.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-166}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (/ (* (* D (* D h)) (/ 0.25 l)) (* (/ d M) (/ d M))))))
   (if (<= (/ h l) -5e-296)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -5e-296) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -5e-296) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))))
	elif (h / l) <= -5e-296:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * Float64(D * h)) * Float64(0.25 / l)) / Float64(Float64(d / M) * Float64(d / M))))));
	elseif (Float64(h / l) <= -5e-296)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	elseif ((h / l) <= -5e-296)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -5e-296], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. times-frac 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in M around 0 50.9%

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

      1. associate-*r/ 50.9%

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

      2. *-commutative 50.9%

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

      3. times-frac 55.6%

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

      4. associate-*r* 51.1%

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

      5. unpow2 51.1%

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

      6. associate-*l* 55.2%

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

      7. unpow2 55.2%

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

      8. unpow2 55.2%

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

   6. Simplified 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. associate-/l* 55.2%

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

      4. *-commutative 55.2%

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

   8. Applied egg-rr 55.2%

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

   10. Simplified 69.6%

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

   if -inf.0 < (/.f64 h l) < -5.0000000000000003e-296

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. times-frac 79.0%

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

   3. Simplified 79.0%

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

   if -5.0000000000000003e-296 < (/.f64 h l)

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. times-frac 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in M around 0 95.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e+306)
   (* w0 (sqrt (- 1.0 (/ (* (* D (* D h)) (/ 0.25 l)) (* (/ d M) (/ d M))))))
   (if (<= (/ h l) -5e-296)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M (/ (* d 2.0) D)) 2.0)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+306) {
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -5e-296) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M / ((d * 2.0) / D)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 ((h / l) <= (-1d+306)) then
        tmp = w0 * sqrt((1.0d0 - (((d * (d * h)) * (0.25d0 / l)) / ((d_1 / m) * (d_1 / m)))))
    else if ((h / l) <= (-5d-296)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m / ((d_1 * 2.0d0) / d)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+306) {
		tmp = w0 * Math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if ((h / l) <= -5e-296) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M / ((d * 2.0) / D)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e+306:
		tmp = w0 * math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))))
	elif (h / l) <= -5e-296:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M / ((d * 2.0) / D)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * Float64(D * h)) * Float64(0.25 / l)) / Float64(Float64(d / M) * Float64(d / M))))));
	elseif (Float64(h / l) <= -5e-296)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M / Float64(Float64(d * 2.0) / D)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+306)
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	elseif ((h / l) <= -5e-296)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M / ((d * 2.0) / D)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+306], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -5e-296], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M / N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1.00000000000000002e306

   1. Initial program 41.6%

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

      1. *-commutative 41.6%

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

      2. times-frac 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in M around 0 49.0%

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

      1. associate-*r/ 49.0%

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

      2. *-commutative 49.0%

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

      3. times-frac 53.5%

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

      4. associate-*r* 49.2%

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

      5. unpow2 49.2%

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

      6. associate-*l* 53.2%

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

      7. unpow2 53.2%

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

      8. unpow2 53.2%

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

   6. Simplified 53.2%

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

      1. *-un-lft-identity 53.2%

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

      2. *-commutative 53.2%

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

      3. associate-/l* 53.2%

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

      4. *-commutative 53.2%

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

   8. Applied egg-rr 53.2%

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

   10. Simplified 67.0%

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

   if -1.00000000000000002e306 < (/.f64 h l) < -5.0000000000000003e-296

   1. Initial program 78.9%

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

      1. associate-/l* 81.2%

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

   3. Simplified 81.2%

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

   if -5.0000000000000003e-296 < (/.f64 h l)

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. times-frac 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in M around 0 95.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l)))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 2.0)) / l)));
}

Fortran

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 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.2%

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

   3. *-commutative 79.2%

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

   4. associate-*l/ 84.8%

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

   5. *-commutative 84.8%

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

   6. frac-times 85.1%

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

   7. div-inv 85.1%

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

   8. metadata-eval 85.1%

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

5. Applied egg-rr 85.1%

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

6. Final simplification 85.1%

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

Alternative 7: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+102}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-161}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d \cdot \ell} \cdot \left(\frac{D}{d} \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M -1e+102)
   (+ w0 (* w0 (* -0.125 (* D (* (/ D l) (* (/ M d) (* M (/ h d))))))))
   (if (<= M -2e-161)
     (* w0 (sqrt (- 1.0 (* (/ 0.25 (* d l)) (* (/ D d) (* D (* h (* M M))))))))
     (if (<= M 7e-203)
       (* w0 (+ 1.0 (* (/ -0.125 l) (* D (* D (/ (* (/ M d) (* M h)) d))))))
       (*
        w0
        (sqrt
         (+ 1.0 (* (* D (* (/ D l) (* (/ (* M M) d) (/ h d)))) -0.25))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1e+102) {
		tmp = w0 + (w0 * (-0.125 * (D * ((D / l) * ((M / d) * (M * (h / d)))))));
	} else if (M <= -2e-161) {
		tmp = w0 * sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	} else if (M <= 7e-203) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Fortran

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 (m <= (-1d+102)) then
        tmp = w0 + (w0 * ((-0.125d0) * (d * ((d / l) * ((m / d_1) * (m * (h / d_1)))))))
    else if (m <= (-2d-161)) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / (d_1 * l)) * ((d / d_1) * (d * (h * (m * m)))))))
    else if (m <= 7d-203) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (d * (d * (((m / d_1) * (m * h)) / d_1)))))
    else
        tmp = w0 * sqrt((1.0d0 + ((d * ((d / l) * (((m * m) / d_1) * (h / d_1)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1e+102) {
		tmp = w0 + (w0 * (-0.125 * (D * ((D / l) * ((M / d) * (M * (h / d)))))));
	} else if (M <= -2e-161) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	} else if (M <= 7e-203) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1e+102:
		tmp = w0 + (w0 * (-0.125 * (D * ((D / l) * ((M / d) * (M * (h / d)))))))
	elif M <= -2e-161:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))))
	elif M <= 7e-203:
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1e+102)
		tmp = Float64(w0 + Float64(w0 * Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(Float64(M / d) * Float64(M * Float64(h / d))))))));
	elseif (M <= -2e-161)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / Float64(d * l)) * Float64(Float64(D / d) * Float64(D * Float64(h * Float64(M * M))))))));
	elseif (M <= 7e-203)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(M * h)) / d))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D * Float64(Float64(D / l) * Float64(Float64(Float64(M * M) / d) * Float64(h / d)))) * -0.25))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -1e+102)
		tmp = w0 + (w0 * (-0.125 * (D * ((D / l) * ((M / d) * (M * (h / d)))))));
	elseif (M <= -2e-161)
		tmp = w0 * sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	elseif (M <= 7e-203)
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	else
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -1e+102], N[(w0 + N[(w0 * N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -2e-161], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 7e-203], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if M < -9.99999999999999977e101

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in M around 0 24.2%

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

      1. associate-*r/ 24.2%

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

      2. *-commutative 24.2%

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

      3. associate-*r/ 24.2%

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

      4. *-commutative 24.2%

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

      5. times-frac 24.0%

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

      6. unpow2 24.0%

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

      7. *-commutative 24.0%

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

      8. unpow2 24.0%

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

      9. unpow2 24.0%

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

   6. Simplified 24.0%

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

   7. Taylor expanded in D around 0 24.2%

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

      1. *-commutative 24.2%

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

      2. times-frac 24.0%

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

      3. unpow2 24.0%

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

      4. unpow2 24.0%

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

      5. times-frac 34.3%

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

      6. unpow2 34.3%

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

      7. associate-*l/ 37.5%

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

      8. *-commutative 37.5%

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

      9. associate-/l* 50.6%

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

   9. Simplified 50.6%

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

       1. associate-*r/ 47.2%

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

       2. associate-/r/ 47.2%

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

   11. Applied egg-rr 47.2%

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

       1. distribute-rgt-in 47.2%

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

       2. *-un-lft-identity 47.2%

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

       3. *-commutative 47.2%

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

       4. associate-*l* 47.6%

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

       5. associate-*r/ 50.8%

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

       6. associate-*l* 50.8%

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

   13. Applied egg-rr 50.8%

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

   if -9.99999999999999977e101 < M < -2.00000000000000006e-161

   1. Initial program 89.9%

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

      1. *-commutative 89.9%

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

      2. times-frac 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in M around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. times-frac 73.3%

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

      4. associate-*r* 68.3%

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

      5. unpow2 68.3%

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

      6. associate-*l* 71.7%

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

      7. unpow2 71.7%

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

      8. unpow2 71.7%

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

   6. Simplified 71.7%

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

      1. associate-*r/ 71.6%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

   8. Applied egg-rr 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. times-frac 90.5%

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

      3. *-commutative 90.5%

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

   10. Applied egg-rr 90.5%

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

       1. *-lft-identity 90.5%

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

       2. associate-/r* 90.5%

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

       3. associate-/l* 92.3%

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

       4. associate-/r/ 92.3%

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

       5. *-commutative 92.3%

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

       6. *-commutative 92.3%

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

       7. associate-*r* 92.2%

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

   12. Simplified 92.2%

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

   if -2.00000000000000006e-161 < M < 7.0000000000000003e-203

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. frac-times 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-*l/ 94.4%

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

      5. *-commutative 94.4%

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

      6. frac-times 95.7%

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

      7. div-inv 95.7%

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

      8. metadata-eval 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in h around 0 68.7%

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

   8. Simplified 77.8%

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

      1. fma-udef 77.8%

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

      2. associate-*l* 86.8%

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

      3. associate-/r/ 86.8%

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

   10. Applied egg-rr 86.8%

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

       1. associate-*r/ 91.3%

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

       2. associate-*l* 92.9%

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

   12. Applied egg-rr 92.9%

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

   if 7.0000000000000003e-203 < M

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. times-frac 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in w0 around 0 53.2%

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

      1. *-commutative 53.2%

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

      2. cancel-sign-sub-inv 53.2%

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

      3. metadata-eval 53.2%

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

      4. *-commutative 53.2%

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

      5. associate-/l* 53.3%

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

      6. associate-*r/ 53.3%

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

      7. unpow2 53.3%

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

      8. associate-/r* 50.5%

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

      9. *-commutative 50.5%

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

      10. unpow2 50.5%

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

      11. associate-*l* 53.9%

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

      12. unpow2 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in D around 0 53.2%

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

      1. associate-*r/ 53.2%

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

      2. unpow2 53.2%

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

      3. *-commutative 53.2%

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

      4. unpow2 53.2%

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

      5. associate-*r/ 53.2%

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

      6. *-commutative 53.2%

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

   9. Simplified 67.8%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+102}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq -2 \cdot 10^{-161}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d \cdot \ell} \cdot \left(\frac{D}{d} \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}\\ \mathbf{elif}\;M \leq 7 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \]

Alternative 8: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+101}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;M \leq -1.4 \cdot 10^{-162}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d \cdot \ell} \cdot \left(\frac{D}{d} \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M -1e+101)
   (* w0 (sqrt (- 1.0 (/ (* (* D (* D h)) (/ 0.25 l)) (* (/ d M) (/ d M))))))
   (if (<= M -1.4e-162)
     (* w0 (sqrt (- 1.0 (* (/ 0.25 (* d l)) (* (/ D d) (* D (* h (* M M))))))))
     (if (<= M 5e-203)
       (* w0 (+ 1.0 (* (/ -0.125 l) (* D (* D (/ (* (/ M d) (* M h)) d))))))
       (*
        w0
        (sqrt
         (+ 1.0 (* (* D (* (/ D l) (* (/ (* M M) d) (/ h d)))) -0.25))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1e+101) {
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if (M <= -1.4e-162) {
		tmp = w0 * sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	} else if (M <= 5e-203) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Fortran

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 (m <= (-1d+101)) then
        tmp = w0 * sqrt((1.0d0 - (((d * (d * h)) * (0.25d0 / l)) / ((d_1 / m) * (d_1 / m)))))
    else if (m <= (-1.4d-162)) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / (d_1 * l)) * ((d / d_1) * (d * (h * (m * m)))))))
    else if (m <= 5d-203) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (d * (d * (((m / d_1) * (m * h)) / d_1)))))
    else
        tmp = w0 * sqrt((1.0d0 + ((d * ((d / l) * (((m * m) / d_1) * (h / d_1)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1e+101) {
		tmp = w0 * Math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	} else if (M <= -1.4e-162) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	} else if (M <= 5e-203) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1e+101:
		tmp = w0 * math.sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))))
	elif M <= -1.4e-162:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))))
	elif M <= 5e-203:
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1e+101)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * Float64(D * h)) * Float64(0.25 / l)) / Float64(Float64(d / M) * Float64(d / M))))));
	elseif (M <= -1.4e-162)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / Float64(d * l)) * Float64(Float64(D / d) * Float64(D * Float64(h * Float64(M * M))))))));
	elseif (M <= 5e-203)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(M * h)) / d))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D * Float64(Float64(D / l) * Float64(Float64(Float64(M * M) / d) * Float64(h / d)))) * -0.25))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -1e+101)
		tmp = w0 * sqrt((1.0 - (((D * (D * h)) * (0.25 / l)) / ((d / M) * (d / M)))));
	elseif (M <= -1.4e-162)
		tmp = w0 * sqrt((1.0 - ((0.25 / (d * l)) * ((D / d) * (D * (h * (M * M)))))));
	elseif (M <= 5e-203)
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	else
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -1e+101], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -1.4e-162], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5e-203], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if M < -9.9999999999999998e100

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in M around 0 24.2%

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

      1. associate-*r/ 24.2%

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

      2. *-commutative 24.2%

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

      3. times-frac 24.2%

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

      4. associate-*r* 33.8%

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

      5. unpow2 33.8%

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

      6. associate-*l* 34.4%

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

      7. unpow2 34.4%

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

      8. unpow2 34.4%

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

   6. Simplified 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. *-commutative 34.4%

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

      3. associate-/l* 37.5%

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

      4. *-commutative 37.5%

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

   8. Applied egg-rr 37.5%

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

   10. Simplified 54.6%

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

   if -9.9999999999999998e100 < M < -1.40000000000000011e-162

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. times-frac 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in M around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. *-commutative 71.8%

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

      3. times-frac 73.7%

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

      4. associate-*r* 68.8%

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

      5. unpow2 68.8%

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

      6. associate-*l* 72.2%

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

      7. unpow2 72.2%

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

      8. unpow2 72.2%

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

   6. Simplified 72.2%

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

      1. associate-*r/ 72.1%

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

      2. associate-*l* 79.1%

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

      3. *-commutative 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. times-frac 90.7%

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

      3. *-commutative 90.7%

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

   10. Applied egg-rr 90.7%

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

       1. *-lft-identity 90.7%

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

       2. associate-/r* 90.7%

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

       3. associate-/l* 92.4%

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

       4. associate-/r/ 92.4%

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

       5. *-commutative 92.4%

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

       6. *-commutative 92.4%

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

       7. associate-*r* 92.4%

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

   12. Simplified 92.4%

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

   if -1.40000000000000011e-162 < M < 5.0000000000000002e-203

   1. Initial program 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 89.7%

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

   3. Simplified 89.7%

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

      1. *-commutative 89.7%

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

      2. frac-times 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*l/ 94.3%

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

      5. *-commutative 94.3%

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

      6. frac-times 95.6%

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

      7. div-inv 95.6%

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

      8. metadata-eval 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in h around 0 68.2%

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

   8. Simplified 77.5%

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

      1. fma-udef 77.5%

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

      2. associate-*l* 86.6%

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

      3. associate-/r/ 86.6%

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

   10. Applied egg-rr 86.6%

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

       1. associate-*r/ 91.2%

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

       2. associate-*l* 92.8%

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

   12. Applied egg-rr 92.8%

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

   if 5.0000000000000002e-203 < M

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. times-frac 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in w0 around 0 53.2%

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

      1. *-commutative 53.2%

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

      2. cancel-sign-sub-inv 53.2%

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

      3. metadata-eval 53.2%

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

      4. *-commutative 53.2%

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

      5. associate-/l* 53.3%

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

      6. associate-*r/ 53.3%

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

      7. unpow2 53.3%

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

      8. associate-/r* 50.5%

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

      9. *-commutative 50.5%

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

      10. unpow2 50.5%

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

      11. associate-*l* 53.9%

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

      12. unpow2 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in D around 0 53.2%

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

      1. associate-*r/ 53.2%

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

      2. unpow2 53.2%

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

      3. *-commutative 53.2%

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

      4. unpow2 53.2%

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

      5. associate-*r/ 53.2%

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

      6. *-commutative 53.2%

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

   9. Simplified 67.8%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+101}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{M} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;M \leq -1.4 \cdot 10^{-162}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d \cdot \ell} \cdot \left(\frac{D}{d} \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \]

Alternative 9: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \leq -1 \cdot 10^{-265}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= w0 -1e-265)
   (* w0 (+ 1.0 (* (/ -0.125 l) (* D (* D (/ (* (/ M d) (* M h)) d))))))
   (*
    w0
    (sqrt (+ 1.0 (* (* D (* (/ D l) (* (/ (* M M) d) (/ h d)))) -0.25))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (w0 <= -1e-265) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Fortran

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 (w0 <= (-1d-265)) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (d * (d * (((m / d_1) * (m * h)) / d_1)))))
    else
        tmp = w0 * sqrt((1.0d0 + ((d * ((d / l) * (((m * m) / d_1) * (h / d_1)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (w0 <= -1e-265) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if w0 <= -1e-265:
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (w0 <= -1e-265)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(M * h)) / d))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D * Float64(Float64(D / l) * Float64(Float64(Float64(M * M) / d) * Float64(h / d)))) * -0.25))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (w0 <= -1e-265)
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
	else
		tmp = w0 * sqrt((1.0 + ((D * ((D / l) * (((M * M) / d) * (h / d)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[w0, -1e-265], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w0 < -9.99999999999999985e-266

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. times-frac 80.7%

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

   3. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. frac-times 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*l/ 87.1%

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

      5. *-commutative 87.1%

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

      6. frac-times 87.9%

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

      7. div-inv 87.9%

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

      8. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

   6. Taylor expanded in h around 0 55.1%

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

   8. Simplified 65.3%

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

      1. fma-udef 65.3%

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

      2. associate-*l* 74.1%

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

      3. associate-/r/ 74.1%

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

   10. Applied egg-rr 74.1%

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

       1. associate-*r/ 75.0%

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

       2. associate-*l* 76.0%

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

   12. Applied egg-rr 76.0%

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

   if -9.99999999999999985e-266 < w0

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

      2. times-frac 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in w0 around 0 58.8%

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

      1. *-commutative 58.8%

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

      2. cancel-sign-sub-inv 58.8%

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

      3. metadata-eval 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-/l* 59.5%

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

      6. associate-*r/ 59.5%

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

      7. unpow2 59.5%

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

      8. associate-/r* 59.0%

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

      9. *-commutative 59.0%

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

      10. unpow2 59.0%

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

      11. associate-*l* 62.6%

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

      12. unpow2 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in D around 0 58.8%

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

      1. associate-*r/ 58.8%

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

      2. unpow2 58.8%

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

      3. *-commutative 58.8%

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

      4. unpow2 58.8%

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

      5. associate-*r/ 58.8%

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

      6. *-commutative 58.8%

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

   9. Simplified 76.8%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \leq -1 \cdot 10^{-265}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot h\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.25}\\ \end{array} \]

Alternative 10: 74.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -4.2 \cdot 10^{-10} \lor \neg \left(M \leq 2.5 \cdot 10^{-59}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (or (<= M -4.2e-10) (not (<= M 2.5e-59)))
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (/ (* M (/ h d)) (/ d M))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M <= -4.2e-10) || !(M <= 2.5e-59)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 ((m <= (-4.2d-10)) .or. (.not. (m <= 2.5d-59))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((m * (h / d_1)) / (d_1 / m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M <= -4.2e-10) || !(M <= 2.5e-59)) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M <= -4.2e-10) or not (M <= 2.5e-59):
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((M <= -4.2e-10) || !(M <= 2.5e-59))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M * Float64(h / d)) / Float64(d / M))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M <= -4.2e-10) || ~((M <= 2.5e-59)))
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[Or[LessEqual[M, -4.2e-10], N[Not[LessEqual[M, 2.5e-59]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -4.2e-10 or 2.5000000000000001e-59 < M

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. times-frac 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in M around 0 42.1%

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

      1. associate-*r/ 42.1%

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

      2. *-commutative 42.1%

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

      3. associate-*r/ 42.1%

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

      4. *-commutative 42.1%

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

      5. times-frac 42.3%

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

      6. unpow2 42.3%

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

      7. *-commutative 42.3%

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

      8. unpow2 42.3%

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

      9. unpow2 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in D around 0 42.1%

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

      1. *-commutative 42.1%

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

      2. times-frac 42.3%

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

      3. unpow2 42.3%

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

      4. unpow2 42.3%

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

      5. times-frac 47.8%

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

      6. unpow2 47.8%

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

      7. associate-*l/ 50.4%

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

      8. *-commutative 50.4%

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

      9. associate-/l* 58.3%

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

   9. Simplified 58.3%

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

       1. associate-*l/ 61.0%

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

   11. Applied egg-rr 61.0%

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

   if -4.2e-10 < M < 2.5000000000000001e-59

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in M around 0 84.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.2 \cdot 10^{-10} \lor \neg \left(M \leq 2.5 \cdot 10^{-59}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 72.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

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 -5.6e-74)
   w0
   (if (<= d 2.9e-8)
     (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (* (/ h d) (/ M (/ d M)))))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -5.6e-74) {
		tmp = w0;
	} else if (d <= 2.9e-8) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h / d) * (M / (d / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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_1 <= (-5.6d-74)) then
        tmp = w0
    else if (d_1 <= 2.9d-8) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((h / d_1) * (m / (d_1 / m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -5.6e-74) {
		tmp = w0;
	} else if (d <= 2.9e-8) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h / d) * (M / (d / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= -5.6e-74:
		tmp = w0
	elif d <= 2.9e-8:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h / d) * (M / (d / M))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= -5.6e-74)
		tmp = w0;
	elseif (d <= 2.9e-8)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h / d) * Float64(M / Float64(d / M)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= -5.6e-74)
		tmp = w0;
	elseif (d <= 2.9e-8)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h / d) * (M / (d / M))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, -5.6e-74], w0, If[LessEqual[d, 2.9e-8], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < -5.59999999999999976e-74 or 2.9000000000000002e-8 < d

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 81.7%

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

   if -5.59999999999999976e-74 < d < 2.9000000000000002e-8

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. times-frac 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in M around 0 40.8%

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

      1. associate-*r/ 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r/ 40.8%

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

      4. *-commutative 40.8%

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

      5. times-frac 43.9%

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

      6. unpow2 43.9%

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

      7. *-commutative 43.9%

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

      8. unpow2 43.9%

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

      9. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in D around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. times-frac 43.9%

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

      3. unpow2 43.9%

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

      4. unpow2 43.9%

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

      5. times-frac 55.7%

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

      6. unpow2 55.7%

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

      7. associate-*l/ 57.9%

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

      8. *-commutative 57.9%

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

      9. associate-/l* 59.1%

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

   9. Simplified 59.1%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)} \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 12: 73.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-73}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \left(M \cdot \frac{M}{d}\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

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 -1.4e-73)
   w0
   (if (<= d 3.1e-9)
     (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (/ (* h (* M (/ M d))) d)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -1.4e-73) {
		tmp = w0;
	} else if (d <= 3.1e-9) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h * (M * (M / d))) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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_1 <= (-1.4d-73)) then
        tmp = w0
    else if (d_1 <= 3.1d-9) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((h * (m * (m / d_1))) / d_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -1.4e-73) {
		tmp = w0;
	} else if (d <= 3.1e-9) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h * (M * (M / d))) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= -1.4e-73:
		tmp = w0
	elif d <= 3.1e-9:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h * (M * (M / d))) / d))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= -1.4e-73)
		tmp = w0;
	elseif (d <= 3.1e-9)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M * Float64(M / d))) / d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= -1.4e-73)
		tmp = w0;
	elseif (d <= 3.1e-9)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((h * (M * (M / d))) / d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, -1.4e-73], w0, If[LessEqual[d, 3.1e-9], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < -1.40000000000000006e-73 or 3.10000000000000005e-9 < d

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 81.7%

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

   if -1.40000000000000006e-73 < d < 3.10000000000000005e-9

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. times-frac 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in M around 0 40.8%

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

      1. associate-*r/ 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-*r/ 40.8%

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

      4. *-commutative 40.8%

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

      5. times-frac 43.9%

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

      6. unpow2 43.9%

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

      7. *-commutative 43.9%

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

      8. unpow2 43.9%

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

      9. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in D around 0 40.8%

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

      1. *-commutative 40.8%

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

      2. times-frac 43.9%

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

      3. unpow2 43.9%

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

      4. unpow2 43.9%

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

      5. times-frac 55.7%

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

      6. unpow2 55.7%

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

      7. associate-*l/ 57.9%

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

      8. *-commutative 57.9%

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

      9. associate-/l* 59.1%

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

   9. Simplified 59.1%

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

       1. associate-*r/ 61.2%

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

       2. associate-/r/ 61.2%

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

   11. Applied egg-rr 61.2%

       \[\leadsto w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot M\right) \cdot h}{d}}\right) \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-73}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \left(M \cdot \frac{M}{d}\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 13: 73.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -7 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right)\right)\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{M}{d} \cdot \left(M \cdot h\right)\right) \cdot \frac{D \cdot D}{\ell}}{d}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M -7e-11)
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (/ (* M (/ h d)) (/ d M))))))
   (if (<= M 5e-59)
     w0
     (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ M d) (* M h)) (/ (* D D) l)) d)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -7e-11) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	} else if (M <= 5e-59) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((M / d) * (M * h)) * ((D * D) / l)) / d)));
	}
	return tmp;
}

Fortran

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 (m <= (-7d-11)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((m * (h / d_1)) / (d_1 / m)))))
    else if (m <= 5d-59) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((m / d_1) * (m * h)) * ((d * d) / l)) / d_1)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -7e-11) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	} else if (M <= 5e-59) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((M / d) * (M * h)) * ((D * D) / l)) / d)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -7e-11:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))))
	elif M <= 5e-59:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((M / d) * (M * h)) * ((D * D) / l)) / d)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -7e-11)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M * Float64(h / d)) / Float64(d / M))))));
	elseif (M <= 5e-59)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(M / d) * Float64(M * h)) * Float64(Float64(D * D) / l)) / d))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -7e-11)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (h / d)) / (d / M)))));
	elseif (M <= 5e-59)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((((M / d) * (M * h)) * ((D * D) / l)) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -7e-11], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5e-59], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(M / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -7.00000000000000038e-11

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

      2. times-frac 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in M around 0 34.1%

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

      1. associate-*r/ 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*r/ 34.1%

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

      4. *-commutative 34.1%

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

      5. times-frac 36.2%

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

      6. unpow2 36.2%

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

      7. *-commutative 36.2%

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

      8. unpow2 36.2%

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

      9. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in D around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. times-frac 36.2%

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

      3. unpow2 36.2%

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

      4. unpow2 36.2%

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

      5. times-frac 43.1%

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

      6. unpow2 43.1%

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

      7. associate-*l/ 47.4%

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

      8. *-commutative 47.4%

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

      9. associate-/l* 56.2%

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

   9. Simplified 56.2%

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

       1. associate-*l/ 56.2%

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

   11. Applied egg-rr 56.2%

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

   if -7.00000000000000038e-11 < M < 5.0000000000000001e-59

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in M around 0 84.0%

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

   if 5.0000000000000001e-59 < M

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. times-frac 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in M around 0 47.5%

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

      1. associate-*r/ 47.5%

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

      2. *-commutative 47.5%

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

      3. associate-*r/ 47.5%

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

      4. *-commutative 47.5%

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

      5. times-frac 46.4%

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

      6. unpow2 46.4%

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

      7. *-commutative 46.4%

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

      8. unpow2 46.4%

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

      9. unpow2 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in D around 0 47.5%

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

      1. *-commutative 47.5%

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

      2. times-frac 46.4%

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

      3. unpow2 46.4%

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

      4. unpow2 46.4%

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

      5. times-frac 50.9%

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

      6. unpow2 50.9%

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

      7. associate-*l/ 52.4%

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

      8. *-commutative 52.4%

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

      9. associate-/l* 59.8%

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

   9. Simplified 59.8%

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

       1. associate-*r/ 56.8%

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

       2. associate-/r/ 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. associate-*r/ 56.9%

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

       2. associate-*r/ 55.5%

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

       3. associate-*l* 59.8%

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

   13. Applied egg-rr 59.8%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \left(M \cdot h\right)\right)}{d}} \cdot -0.125\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}\right)\right)\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{M}{d} \cdot \left(M \cdot h\right)\right) \cdot \frac{D \cdot D}{\ell}}{d}\right)\\ \end{array} \]

Alternative 14: 74.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= l 5.8e+69)
   (* w0 (+ 1.0 (* (/ -0.125 l) (* D (* D (* (/ h d) (* M (/ M d))))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5.8e+69) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * ((h / d) * (M * (M / d)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 (l <= 5.8d+69) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (d * (d * ((h / d_1) * (m * (m / d_1)))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5.8e+69) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * ((h / d) * (M * (M / d)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if l <= 5.8e+69:
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * ((h / d) * (M * (M / d)))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (l <= 5.8e+69)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D * Float64(D * Float64(Float64(h / d) * Float64(M * Float64(M / d))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (l <= 5.8e+69)
		tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * ((h / d) * (M * (M / d)))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[l, 5.8e+69], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D * N[(D * N[(N[(h / d), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if l < 5.7999999999999997e69

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

      1. *-commutative 79.8%

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

      2. frac-times 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*l/ 86.3%

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

      5. *-commutative 86.3%

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

      6. frac-times 87.1%

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

      7. div-inv 87.1%

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

      8. metadata-eval 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in h around 0 57.8%

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

   8. Simplified 69.3%

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

      1. fma-udef 69.3%

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

      2. associate-*l* 75.0%

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

      3. associate-/r/ 75.0%

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

   10. Applied egg-rr 75.0%

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

   if 5.7999999999999997e69 < l

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. times-frac 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 78.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 15: 75.5% accurate, 10.3× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* (/ -0.125 l) (* D (* D (/ (* (/ M d) (* M h)) d)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
}

Fortran

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 * (1.0d0 + (((-0.125d0) / l) * (d * (d * (((m / d_1) * (m * h)) / d_1)))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D * Float64(D * Float64(Float64(Float64(M / d) * Float64(M * h)) / d))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + ((-0.125 / l) * (D * (D * (((M / d) * (M * h)) / d)))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. *-commutative 79.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.2%

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

   3. *-commutative 79.2%

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

   4. associate-*l/ 84.8%

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

   5. *-commutative 84.8%

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

   6. frac-times 85.1%

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

   7. div-inv 85.1%

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

   8. metadata-eval 85.1%

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

5. Applied egg-rr 85.1%

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

6. Taylor expanded in h around 0 55.0%

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

8. Simplified 67.1%

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

   1. fma-udef 67.1%

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

   2. associate-*l* 73.7%

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

   3. associate-/r/ 73.7%

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

10. Applied egg-rr 73.7%

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

    1. associate-*r/ 73.7%

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

    2. associate-*l* 74.2%

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

12. Applied egg-rr 74.2%

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

13. Final simplification 74.2%

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

Alternative 16: 67.6% accurate, 216.0× speedup

Math

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

FPCore

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

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 79.2%

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

   1. *-commutative 79.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in M around 0 68.3%

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

5. Final simplification 68.3%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023185 
(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))))))