Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 88.0%

Time: 15.9s

Alternatives: 4

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: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M\_m \cdot \frac{D\_m}{d}\right)\\ \mathbf{if}\;M\_m \leq 2 \cdot 10^{-43}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M_m (/ D_m d)))))
   (if (<= M_m 2e-43)
     (* w0 (sqrt (- 1.0 (/ (* (pow (/ (* M_m (* D_m 0.5)) d) 2.0) h) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (M_m <= 2e-43) {
		tmp = w0 * sqrt((1.0 - ((pow(((M_m * (D_m * 0.5)) / d), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (m_m * (d_m / d))
    if (m_m <= 2d-43) then
        tmp = w0 * sqrt((1.0d0 - (((((m_m * (d_m * 0.5d0)) / d) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (M_m <= 2e-43) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow(((M_m * (D_m * 0.5)) / d), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (M_m * (D_m / d))
	tmp = 0
	if M_m <= 2e-43:
		tmp = w0 * math.sqrt((1.0 - ((math.pow(((M_m * (D_m * 0.5)) / d), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(M_m * Float64(D_m / d)))
	tmp = 0.0
	if (M_m <= 2e-43)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (M_m * (D_m / d));
	tmp = 0.0;
	if (M_m <= 2e-43)
		tmp = w0 * sqrt((1.0 - (((((M_m * (D_m * 0.5)) / d) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 2e-43], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.00000000000000015e-43

   1. Initial program 81.5%

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

   3. Simplified 81.0%

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

      1. associate-*r/ 81.5%

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

      2. associate-/r* 81.5%

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

      3. add-sqr-sqrt 43.6%

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

      4. associate-/r* 43.6%

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

      5. associate-/l* 43.6%

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

      6. div-inv 43.6%

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

      7. metadata-eval 43.6%

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

   6. Applied egg-rr 43.6%

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

      1. associate-*r/ 46.4%

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

      2. associate-/l/ 46.4%

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

      3. add-sqr-sqrt 86.1%

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

   8. Applied egg-rr 86.1%

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

   if 2.00000000000000015e-43 < M

   1. Initial program 77.8%

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

   3. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. un-div-inv 80.3%

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

      3. associate-/l* 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. clear-num 80.3%

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

      2. inv-pow 80.3%

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

   8. Applied egg-rr 80.3%

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

      1. associate-*r/ 84.2%

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

      2. pow-pow 84.2%

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

      3. associate-/l* 84.2%

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

      4. unpow-prod-down 84.2%

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

      5. metadata-eval 84.2%

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

      6. metadata-eval 84.2%

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

      7. metadata-eval 84.2%

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

   10. Applied egg-rr 84.2%

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

       1. associate-/l* 80.3%

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

       2. add-sqr-sqrt 80.3%

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

       3. associate-*l* 80.3%

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

       4. sqrt-prod 80.3%

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

       5. metadata-eval 80.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{0.5} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       6. sqrt-pow1 64.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{{\left(\frac{\frac{d}{D}}{M}\right)}^{\left(\frac{-2}{2}\right)}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       7. metadata-eval 64.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{\color{blue}{-1}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       8. unpow-1 64.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{d}{D}}{M}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       9. clear-num 64.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       10. div-inv 64.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       11. clear-num 64.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \color{blue}{\frac{D}{d}}\right)\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       12. sqrt-prod 64.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right)} \cdot \frac{h}{\ell}\right)} \]

       13. metadata-eval 64.4%

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

       14. sqrt-pow1 85.2%

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

       15. metadata-eval 85.2%

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

       16. unpow-1 85.2%

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

       17. clear-num 85.1%

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

       18. div-inv 85.1%

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

       19. clear-num 85.2%

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

   12. Applied egg-rr 85.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-43}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M\_m \cdot \frac{D\_m}{d}\right)\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \leq 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M\_m}\right)}^{-2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M_m (/ D_m d)))))
   (if (<= (pow (/ (* M_m D_m) (* d 2.0)) 2.0) 1e-19)
     (* w0 (sqrt (- 1.0 (/ (* h (* 0.25 (pow (/ (/ d D_m) M_m) -2.0))) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (pow(((M_m * D_m) / (d * 2.0)), 2.0) <= 1e-19) {
		tmp = w0 * sqrt((1.0 - ((h * (0.25 * pow(((d / D_m) / M_m), -2.0))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (m_m * (d_m / d))
    if ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) <= 1d-19) then
        tmp = w0 * sqrt((1.0d0 - ((h * (0.25d0 * (((d / d_m) / m_m) ** (-2.0d0)))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) <= 1e-19) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (0.25 * Math.pow(((d / D_m) / M_m), -2.0))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (M_m * (D_m / d))
	tmp = 0
	if math.pow(((M_m * D_m) / (d * 2.0)), 2.0) <= 1e-19:
		tmp = w0 * math.sqrt((1.0 - ((h * (0.25 * math.pow(((d / D_m) / M_m), -2.0))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(M_m * Float64(D_m / d)))
	tmp = 0.0
	if ((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) <= 1e-19)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(0.25 * (Float64(Float64(d / D_m) / M_m) ^ -2.0))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (M_m * (D_m / d));
	tmp = 0.0;
	if ((((M_m * D_m) / (d * 2.0)) ^ 2.0) <= 1e-19)
		tmp = w0 * sqrt((1.0 - ((h * (0.25 * (((d / D_m) / M_m) ^ -2.0))) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e-19], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.9999999999999998e-20

   1. Initial program 90.7%

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

   3. Simplified 90.7%

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

      1. clear-num 90.7%

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

      2. un-div-inv 90.7%

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

      3. associate-/l* 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. clear-num 90.7%

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

      2. inv-pow 90.7%

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

   8. Applied egg-rr 90.7%

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

      1. associate-*r/ 98.7%

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

      2. pow-pow 98.8%

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

      3. associate-/l* 98.8%

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

      4. unpow-prod-down 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. metadata-eval 98.8%

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

   10. Applied egg-rr 98.8%

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

   if 9.9999999999999998e-20 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 63.3%

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

   3. Simplified 64.2%

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

      1. clear-num 64.2%

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

      2. un-div-inv 64.2%

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

      3. associate-/l* 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. clear-num 64.2%

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

      2. inv-pow 64.2%

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

   8. Applied egg-rr 64.2%

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

      1. associate-*r/ 62.3%

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

      2. pow-pow 62.4%

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

      3. associate-/l* 62.4%

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

      4. unpow-prod-down 62.4%

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

      5. metadata-eval 62.4%

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

      6. metadata-eval 62.4%

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

      7. metadata-eval 62.4%

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

   10. Applied egg-rr 62.4%

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

       1. associate-/l* 64.2%

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

       2. add-sqr-sqrt 64.2%

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

       3. associate-*l* 64.2%

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

       4. sqrt-prod 64.2%

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

       5. metadata-eval 64.2%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{0.5} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       6. sqrt-pow1 40.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{{\left(\frac{\frac{d}{D}}{M}\right)}^{\left(\frac{-2}{2}\right)}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       7. metadata-eval 40.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{\color{blue}{-1}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       8. unpow-1 40.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{d}{D}}{M}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       9. clear-num 40.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       10. div-inv 40.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       11. clear-num 40.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \color{blue}{\frac{D}{d}}\right)\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       12. sqrt-prod 40.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right)} \cdot \frac{h}{\ell}\right)} \]

       13. metadata-eval 40.4%

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

       14. sqrt-pow1 71.2%

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

       15. metadata-eval 71.2%

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

       16. unpow-1 71.2%

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

       17. clear-num 71.2%

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

       18. div-inv 71.1%

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

       19. clear-num 71.2%

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

   12. Applied egg-rr 71.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M\_m \cdot \frac{D\_m}{d}\right)\\ \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M_m (/ D_m d)))))
   (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* d 2.0)) 2.0)) -2e-12)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (d * 2.0)), 2.0)) <= -2e-12) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (m_m * (d_m / d))
    if (((h / l) * (((m_m * d_m) / (d * 2.0d0)) ** 2.0d0)) <= (-2d-12)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (M_m * (D_m / d));
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (d * 2.0)), 2.0)) <= -2e-12) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (M_m * (D_m / d))
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (d * 2.0)), 2.0)) <= -2e-12:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(M_m * Float64(D_m / d)))
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0)) <= -2e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (M_m * (D_m / d));
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (d * 2.0)) ^ 2.0)) <= -2e-12)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $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)) < -1.99999999999999996e-12

   1. Initial program 56.7%

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

   3. Simplified 59.0%

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

      1. clear-num 58.9%

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

      2. un-div-inv 59.0%

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

      3. associate-/l* 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. clear-num 58.9%

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

      2. inv-pow 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*r/ 60.1%

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

      2. pow-pow 60.2%

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

      3. associate-/l* 60.2%

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

      4. unpow-prod-down 60.2%

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

      5. metadata-eval 60.2%

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

      6. metadata-eval 60.2%

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

      7. metadata-eval 60.2%

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

   10. Applied egg-rr 60.2%

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

       1. associate-/l* 58.9%

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

       2. add-sqr-sqrt 58.9%

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

       3. associate-*l* 58.9%

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

       4. sqrt-prod 58.9%

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

       5. metadata-eval 58.9%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{0.5} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       6. sqrt-pow1 27.5%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{{\left(\frac{\frac{d}{D}}{M}\right)}^{\left(\frac{-2}{2}\right)}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       7. metadata-eval 27.5%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{\color{blue}{-1}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       8. unpow-1 27.5%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{d}{D}}{M}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       9. clear-num 27.5%

          \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       10. div-inv 27.5%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \color{blue}{\left(M \cdot \frac{1}{\frac{d}{D}}\right)}\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       11. clear-num 27.5%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \color{blue}{\frac{D}{d}}\right)\right) \cdot \left(\sqrt{0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}} \cdot \frac{h}{\ell}\right)} \]

       12. sqrt-prod 27.5%

           \[\leadsto w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{0.25} \cdot \sqrt{{\left(\frac{\frac{d}{D}}{M}\right)}^{-2}}\right)} \cdot \frac{h}{\ell}\right)} \]

       13. metadata-eval 27.5%

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

       14. sqrt-pow1 65.6%

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

       15. metadata-eval 65.6%

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

       16. unpow-1 65.6%

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

       17. clear-num 65.6%

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

       18. div-inv 65.5%

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

       19. clear-num 65.6%

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

   12. Applied egg-rr 65.6%

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

   if -1.99999999999999996e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 92.4%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in M around 0 98.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Simplified 80.8%

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

5. Taylor expanded in M around 0 67.5%

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

6. Final simplification 67.5%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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))))))