Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 86.7%

Time: 38.9s

Alternatives: 11

Speedup: 1.7×

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: 80.9% 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: 86.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{2 \cdot \frac{\frac{d}{D\_m}}{M\_m}} \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{0.5 \cdot \frac{D\_m \cdot \left(h \cdot M\_m\right)}{\ell \cdot d}}{\frac{1}{D\_m \cdot \frac{M\_m}{d}} \cdot -2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-251)
   (*
    w0
    (sqrt
     (+
      1.0
      (*
       (/ (* h (* M_m (* D_m (/ 0.5 d)))) (* 2.0 (/ (/ d D_m) M_m)))
       (/ -1.0 l)))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/
       (* 0.5 (/ (* D_m (* h M_m)) (* l d)))
       (* (/ 1.0 (* D_m (/ M_m d))) -2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-251) {
		tmp = w0 * sqrt((1.0 + (((h * (M_m * (D_m * (0.5 / d)))) / (2.0 * ((d / D_m) / M_m))) * (-1.0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 + ((0.5 * ((D_m * (h * M_m)) / (l * d))) / ((1.0 / (D_m * (M_m / d))) * -2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-5d-251)) then
        tmp = w0 * sqrt((1.0d0 + (((h * (m_m * (d_m * (0.5d0 / d)))) / (2.0d0 * ((d / d_m) / m_m))) * ((-1.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((0.5d0 * ((d_m * (h * m_m)) / (l * d))) / ((1.0d0 / (d_m * (m_m / d))) * (-2.0d0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-251) {
		tmp = w0 * Math.sqrt((1.0 + (((h * (M_m * (D_m * (0.5 / d)))) / (2.0 * ((d / D_m) / M_m))) * (-1.0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((0.5 * ((D_m * (h * M_m)) / (l * d))) / ((1.0 / (D_m * (M_m / d))) * -2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e-251:
		tmp = w0 * math.sqrt((1.0 + (((h * (M_m * (D_m * (0.5 / d)))) / (2.0 * ((d / D_m) / M_m))) * (-1.0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((0.5 * ((D_m * (h * M_m)) / (l * d))) / ((1.0 / (D_m * (M_m / d))) * -2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-251)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(2.0 * Float64(Float64(d / D_m) / M_m))) * Float64(-1.0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(0.5 * Float64(Float64(D_m * Float64(h * M_m)) / Float64(l * d))) / Float64(Float64(1.0 / Float64(D_m * Float64(M_m / d))) * -2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-251)
		tmp = w0 * sqrt((1.0 + (((h * (M_m * (D_m * (0.5 / d)))) / (2.0 * ((d / D_m) / M_m))) * (-1.0 / l))));
	else
		tmp = w0 * sqrt((1.0 + ((0.5 * ((D_m * (h * M_m)) / (l * d))) / ((1.0 / (D_m * (M_m / d))) * -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-251], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(d / D$95$m), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(0.5 * N[(N[(D$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000003e-251

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-*l/ 80.8%

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

      4. associate-*r/ 80.2%

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

      5. times-frac 80.8%

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

      6. clear-num 80.8%

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

      7. un-div-inv 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-*l/ 80.8%

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

      10. associate-*r/ 80.1%

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

      11. times-frac 80.8%

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

      12. associate-/l* 80.1%

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

      13. div-inv 80.2%

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

      14. associate-/r* 80.2%

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

      15. metadata-eval 80.2%

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

      16. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. associate-*r/ 87.6%

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

      2. associate-*r* 87.6%

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

      3. frac-times 88.2%

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

   8. Applied egg-rr 88.2%

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

      1. div-inv 88.2%

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

      2. associate-*r* 87.6%

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

      3. associate-*l/ 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-/l* 88.3%

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

      6. associate-/r* 88.9%

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

   10. Applied egg-rr 88.9%

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

   if -5.0000000000000003e-251 < (/.f64 h l)

   1. Initial program 87.5%

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

   3. Simplified 88.3%

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

      1. unpow2 88.3%

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

      2. *-commutative 88.3%

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

      3. associate-*l/ 87.5%

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

      4. associate-*r/ 86.5%

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

      5. times-frac 87.5%

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

      6. clear-num 87.5%

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

      7. un-div-inv 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-*l/ 87.5%

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

      10. associate-*r/ 85.7%

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

      11. times-frac 87.5%

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

      12. associate-/l* 85.7%

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

      13. div-inv 85.7%

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

      14. associate-/r* 85.7%

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

      15. metadata-eval 85.7%

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

      16. times-frac 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. associate-*r* 86.6%

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

      3. frac-times 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. *-lft-identity 87.5%

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

      2. sub-neg 87.5%

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

      3. associate-*l/ 92.0%

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

      4. distribute-neg-frac2 92.0%

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

      5. *-commutative 92.0%

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

      6. associate-*l* 89.3%

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

      7. associate-/l* 89.3%

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

      8. *-commutative 89.3%

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

      9. *-commutative 89.3%

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

      10. distribute-rgt-neg-in 89.3%

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

      11. metadata-eval 89.3%

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

   10. Simplified 89.3%

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

       1. clear-num 89.3%

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

       2. inv-pow 89.3%

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

       3. *-commutative 89.3%

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

   12. Applied egg-rr 89.3%

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

       1. unpow-1 89.3%

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

       2. *-commutative 89.3%

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

       3. associate-/l* 91.9%

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

   14. Simplified 91.9%

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

   15. Taylor expanded in h around 0 95.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{2 \cdot \frac{\frac{d}{D}}{M}} \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{0.5 \cdot \frac{D \cdot \left(h \cdot M\right)}{\ell \cdot d}}{\frac{1}{D \cdot \frac{M}{d}} \cdot -2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot 0.5\right)\right)}{d}}{\ell \cdot \frac{d \cdot 2}{M\_m \cdot D\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D\_m}}{\frac{M\_m}{d}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 1e+176)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (/ (* h (* M_m (* D_m 0.5))) d) (* l (/ (* d 2.0) (* M_m D_m)))))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (* (/ h l) (* M_m (* D_m (/ 0.5 d)))) (/ (/ -2.0 D_m) (/ M_m d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e+176) {
		tmp = w0 * sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 1d+176) then
        tmp = w0 * sqrt((1.0d0 - (((h * (m_m * (d_m * 0.5d0))) / d) / (l * ((d * 2.0d0) / (m_m * d_m))))))
    else
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (m_m * (d_m * (0.5d0 / d)))) / (((-2.0d0) / d_m) / (m_m / d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e+176) {
		tmp = w0 * Math.sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 1e+176:
		tmp = w0 * math.sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / d) / (l * ((d * 2.0) / (M_m * D_m))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e+176)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(M_m * Float64(D_m * 0.5))) / d) / Float64(l * Float64(Float64(d * 2.0) / Float64(M_m * D_m)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(Float64(-2.0 / D_m) / Float64(M_m / d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 1e+176)
		tmp = w0 * sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	else
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+176], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[(l * N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / D$95$m), $MachinePrecision] / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e176

   1. Initial program 84.1%

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

   3. Simplified 84.5%

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

      1. unpow2 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-*l/ 84.1%

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

      4. associate-*r/ 83.3%

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

      5. times-frac 84.1%

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

      6. clear-num 84.1%

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

      7. un-div-inv 84.1%

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

      8. *-commutative 84.1%

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

      9. associate-*l/ 84.1%

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

      10. associate-*r/ 82.8%

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

      11. times-frac 84.1%

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

      12. associate-/l* 82.8%

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

      13. div-inv 82.8%

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

      14. associate-/r* 82.8%

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

      15. metadata-eval 82.8%

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

      16. times-frac 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. frac-times 91.4%

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

      2. associate-*r* 90.9%

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

      3. frac-times 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r/ 92.2%

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

      3. associate-*r* 92.2%

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

      4. associate-*r/ 91.6%

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

      5. *-commutative 91.6%

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

      6. *-commutative 91.6%

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

   10. Simplified 91.6%

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

   if 1e176 < (*.f64 M D)

   1. Initial program 79.3%

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

   3. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*l/ 79.3%

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

      4. associate-*r/ 79.3%

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

      5. times-frac 79.3%

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

      6. clear-num 79.3%

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

      7. un-div-inv 79.3%

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

      8. *-commutative 79.3%

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

      9. associate-*l/ 79.3%

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

      10. associate-*r/ 79.3%

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

      11. times-frac 79.3%

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

      12. associate-/l* 79.3%

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

      13. div-inv 79.3%

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

      14. associate-/r* 79.3%

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

      15. metadata-eval 79.3%

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

      16. times-frac 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. associate-*r* 79.3%

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

      3. frac-times 79.3%

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

   8. Applied egg-rr 79.3%

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

      1. *-lft-identity 79.3%

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

      2. sub-neg 79.3%

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

      3. associate-*l/ 79.3%

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

      4. distribute-neg-frac2 79.3%

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

      5. *-commutative 79.3%

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

      6. associate-*l* 79.3%

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

      7. associate-/l* 79.3%

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

      8. *-commutative 79.3%

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

      9. *-commutative 79.3%

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

      10. distribute-rgt-neg-in 79.3%

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

      11. metadata-eval 79.3%

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

   10. Simplified 79.3%

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

       1. clear-num 79.3%

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

       2. inv-pow 79.3%

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

       3. *-commutative 79.3%

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

   12. Applied egg-rr 79.3%

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

       1. unpow-1 79.3%

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

       2. *-commutative 79.3%

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

       3. associate-/l* 91.6%

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

   14. Simplified 91.6%

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

   15. Taylor expanded in D around 0 79.3%

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

       1. associate-*r/ 79.3%

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

       2. *-commutative 79.3%

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

       3. associate-*l/ 79.3%

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

       4. associate-/r/ 79.3%

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

       5. *-commutative 79.3%

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

       6. associate-/l* 91.6%

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

       7. associate-/r* 91.7%

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

   17. Simplified 91.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{+176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{d}}{\ell \cdot \frac{d \cdot 2}{M \cdot D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D}}{\frac{M}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\frac{0.5}{d} \cdot \left(M\_m \cdot D\_m\right)}{\frac{d \cdot 2}{M\_m \cdot D\_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D\_m}}{\frac{M\_m}{d}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 5e-9)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* h (/ (* (/ 0.5 d) (* M_m D_m)) (/ (* d 2.0) (* M_m D_m)))) l))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (* (/ h l) (* M_m (* D_m (/ 0.5 d)))) (/ (/ -2.0 D_m) (/ M_m d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e-9) {
		tmp = w0 * sqrt((1.0 - ((h * (((0.5 / d) * (M_m * D_m)) / ((d * 2.0) / (M_m * D_m)))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 5d-9) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((0.5d0 / d) * (m_m * d_m)) / ((d * 2.0d0) / (m_m * d_m)))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (m_m * (d_m * (0.5d0 / d)))) / (((-2.0d0) / d_m) / (m_m / d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e-9) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (((0.5 / d) * (M_m * D_m)) / ((d * 2.0) / (M_m * D_m)))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 5e-9:
		tmp = w0 * math.sqrt((1.0 - ((h * (((0.5 / d) * (M_m * D_m)) / ((d * 2.0) / (M_m * D_m)))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(Float64(0.5 / d) * Float64(M_m * D_m)) / Float64(Float64(d * 2.0) / Float64(M_m * D_m)))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(Float64(-2.0 / D_m) / Float64(M_m / d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 5e-9)
		tmp = w0 * sqrt((1.0 - ((h * (((0.5 / d) * (M_m * D_m)) / ((d * 2.0) / (M_m * D_m)))) / l)));
	else
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-9], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(N[(0.5 / d), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / D$95$m), $MachinePrecision] / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000001e-9

   1. Initial program 84.9%

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

   3. Simplified 85.4%

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

      1. unpow2 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*l/ 84.9%

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

      4. associate-*r/ 84.0%

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

      5. times-frac 84.9%

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

      6. clear-num 84.9%

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

      7. un-div-inv 84.9%

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

      8. *-commutative 84.9%

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

      9. associate-*l/ 84.9%

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

      10. associate-*r/ 83.5%

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

      11. times-frac 84.9%

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

      12. associate-/l* 83.5%

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

      13. div-inv 83.5%

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

      14. associate-/r* 83.5%

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

      15. metadata-eval 83.5%

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

      16. times-frac 84.0%

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

   6. Applied egg-rr 84.0%

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

      1. associate-*r/ 90.7%

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

      2. associate-*r* 90.7%

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

      3. frac-times 91.6%

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

   8. Applied egg-rr 91.6%

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

   if 5.0000000000000001e-9 < (*.f64 M D)

   1. Initial program 78.1%

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

   3. Simplified 78.1%

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

      1. unpow2 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-*l/ 78.1%

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

      4. associate-*r/ 78.1%

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

      5. times-frac 78.1%

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

      6. clear-num 78.1%

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

      7. un-div-inv 78.1%

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

      8. *-commutative 78.1%

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

      9. associate-*l/ 78.1%

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

      10. associate-*r/ 78.1%

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

      11. times-frac 78.1%

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

      12. associate-/l* 78.1%

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

      13. div-inv 78.1%

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

      14. associate-/r* 78.1%

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

      15. metadata-eval 78.1%

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

      16. times-frac 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. associate-*r* 78.1%

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

      3. frac-times 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. *-lft-identity 78.1%

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

      2. sub-neg 78.1%

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

      3. associate-*l/ 78.1%

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

      4. distribute-neg-frac2 78.1%

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

      5. *-commutative 78.1%

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

      6. associate-*l* 78.1%

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

      7. associate-/l* 78.1%

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

      8. *-commutative 78.1%

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

      9. *-commutative 78.1%

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

      10. distribute-rgt-neg-in 78.1%

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

      11. metadata-eval 78.1%

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

   10. Simplified 78.1%

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

       1. clear-num 78.1%

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

       2. inv-pow 78.1%

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

       3. *-commutative 78.1%

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

   12. Applied egg-rr 78.1%

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

       1. unpow-1 78.1%

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

       2. *-commutative 78.1%

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

       3. associate-/l* 84.0%

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

   14. Simplified 84.0%

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

   15. Taylor expanded in D around 0 78.1%

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

       1. associate-*r/ 78.1%

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

       2. *-commutative 78.1%

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

       3. associate-*l/ 78.1%

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

       4. associate-/r/ 78.1%

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

       5. *-commutative 78.1%

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

       6. associate-/l* 84.0%

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

       7. associate-/r* 84.0%

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

   17. Simplified 84.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\frac{0.5}{d} \cdot \left(M \cdot D\right)}{\frac{d \cdot 2}{M \cdot D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D}}{\frac{M}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-140}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D\_m}}{\frac{M\_m}{d}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-140)
   w0
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (* (/ h l) (* M_m (* D_m (/ 0.5 d)))) (/ (/ -2.0 D_m) (/ M_m d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-140) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 4d-140) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (m_m * (d_m * (0.5d0 / d)))) / (((-2.0d0) / d_m) / (m_m / d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-140) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 4e-140:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-140)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(Float64(-2.0 / D_m) / Float64(M_m / d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 4e-140)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-140], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / D$95$m), $MachinePrecision] / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.9999999999999999e-140

   1. Initial program 83.7%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in D around 0 80.1%

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

   if 3.9999999999999999e-140 < (*.f64 M D)

   1. Initial program 83.6%

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

   3. Simplified 83.5%

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

      1. unpow2 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*l/ 83.6%

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

      4. associate-*r/ 82.1%

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

      5. times-frac 83.6%

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

      6. clear-num 83.6%

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

      7. un-div-inv 83.6%

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

      8. *-commutative 83.6%

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

      9. associate-*l/ 83.6%

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

      10. associate-*r/ 82.1%

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

      11. times-frac 83.6%

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

      12. associate-/l* 82.1%

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

      13. div-inv 82.1%

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

      14. associate-/r* 82.1%

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

      15. metadata-eval 82.1%

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

      16. times-frac 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. associate-*r* 83.6%

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

      3. frac-times 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. *-lft-identity 83.6%

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

      2. sub-neg 83.6%

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

      3. associate-*l/ 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 83.5%

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

      7. associate-/l* 83.5%

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

      8. *-commutative 83.5%

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

      9. *-commutative 83.5%

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

      10. distribute-rgt-neg-in 83.5%

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

      11. metadata-eval 83.5%

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

   10. Simplified 83.5%

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

       1. clear-num 83.6%

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

       2. inv-pow 83.6%

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

       3. *-commutative 83.6%

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

   12. Applied egg-rr 83.6%

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

       1. unpow-1 83.6%

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

       2. *-commutative 83.6%

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

       3. associate-/l* 86.2%

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

   14. Simplified 86.2%

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

   15. Taylor expanded in D around 0 83.5%

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

       1. associate-*r/ 83.5%

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

       2. *-commutative 83.5%

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

       3. associate-*l/ 83.5%

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

       4. associate-/r/ 83.6%

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

       5. *-commutative 83.6%

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

       6. associate-/l* 86.2%

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

       7. associate-/r* 86.2%

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

   17. Simplified 86.2%

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

Alternative 5: 83.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-140}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{-2 \cdot \frac{d}{M\_m \cdot D\_m}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-140)
   w0
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (* (/ h l) (* M_m (* D_m (/ 0.5 d)))) (* -2.0 (/ d (* M_m D_m)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-140) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / (-2.0 * (d / (M_m * D_m))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 4d-140) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (m_m * (d_m * (0.5d0 / d)))) / ((-2.0d0) * (d / (m_m * d_m))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-140) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / (-2.0 * (d / (M_m * D_m))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 4e-140:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / (-2.0 * (d / (M_m * D_m))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-140)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(-2.0 * Float64(d / Float64(M_m * D_m)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 4e-140)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / (-2.0 * (d / (M_m * D_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-140], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.9999999999999999e-140

   1. Initial program 83.7%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in D around 0 80.1%

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

   if 3.9999999999999999e-140 < (*.f64 M D)

   1. Initial program 83.6%

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

   3. Simplified 83.5%

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

      1. unpow2 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*l/ 83.6%

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

      4. associate-*r/ 82.1%

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

      5. times-frac 83.6%

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

      6. clear-num 83.6%

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

      7. un-div-inv 83.6%

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

      8. *-commutative 83.6%

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

      9. associate-*l/ 83.6%

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

      10. associate-*r/ 82.1%

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

      11. times-frac 83.6%

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

      12. associate-/l* 82.1%

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

      13. div-inv 82.1%

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

      14. associate-/r* 82.1%

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

      15. metadata-eval 82.1%

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

      16. times-frac 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. associate-*r* 83.6%

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

      3. frac-times 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. *-lft-identity 83.6%

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

      2. sub-neg 83.6%

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

      3. associate-*l/ 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 83.5%

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

      7. associate-/l* 83.5%

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

      8. *-commutative 83.5%

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

      9. *-commutative 83.5%

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

      10. distribute-rgt-neg-in 83.5%

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

      11. metadata-eval 83.5%

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

   10. Simplified 83.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-140}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{-2 \cdot \frac{d}{M \cdot D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 0.00016:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(0.5 \cdot \left(M\_m \cdot D\_m\right)\right)}{\ell \cdot d}}{-2 \cdot \frac{d}{M\_m \cdot D\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D\_m}}{\frac{M\_m}{d}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 0.00016)
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (/ (* h (* 0.5 (* M_m D_m))) (* l d)) (* -2.0 (/ d (* M_m D_m)))))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/ (* (/ h l) (* M_m (* D_m (/ 0.5 d)))) (/ (/ -2.0 D_m) (/ M_m d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 0.00016) {
		tmp = w0 * sqrt((1.0 + (((h * (0.5 * (M_m * D_m))) / (l * d)) / (-2.0 * (d / (M_m * D_m))))));
	} else {
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 0.00016d0) then
        tmp = w0 * sqrt((1.0d0 + (((h * (0.5d0 * (m_m * d_m))) / (l * d)) / ((-2.0d0) * (d / (m_m * d_m))))))
    else
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (m_m * (d_m * (0.5d0 / d)))) / (((-2.0d0) / d_m) / (m_m / d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 0.00016) {
		tmp = w0 * Math.sqrt((1.0 + (((h * (0.5 * (M_m * D_m))) / (l * d)) / (-2.0 * (d / (M_m * D_m))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 0.00016:
		tmp = w0 * math.sqrt((1.0 + (((h * (0.5 * (M_m * D_m))) / (l * d)) / (-2.0 * (d / (M_m * D_m))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 0.00016)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(0.5 * Float64(M_m * D_m))) / Float64(l * d)) / Float64(-2.0 * Float64(d / Float64(M_m * D_m)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * Float64(0.5 / d)))) / Float64(Float64(-2.0 / D_m) / Float64(M_m / d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 0.00016)
		tmp = w0 * sqrt((1.0 + (((h * (0.5 * (M_m * D_m))) / (l * d)) / (-2.0 * (d / (M_m * D_m))))));
	else
		tmp = w0 * sqrt((1.0 + (((h / l) * (M_m * (D_m * (0.5 / d)))) / ((-2.0 / D_m) / (M_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 0.00016], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 / D$95$m), $MachinePrecision] / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.60000000000000013e-4

   1. Initial program 85.4%

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

   3. Simplified 85.4%

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

      1. unpow2 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*l/ 85.4%

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

      4. associate-*r/ 84.4%

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

      5. times-frac 85.4%

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

      6. clear-num 85.4%

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

      7. un-div-inv 85.4%

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

      8. *-commutative 85.4%

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

      9. associate-*l/ 85.4%

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

      10. associate-*r/ 84.4%

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

      11. times-frac 85.4%

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

      12. associate-/l* 84.4%

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

      13. div-inv 84.4%

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

      14. associate-/r* 84.4%

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

      15. metadata-eval 84.4%

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

      16. times-frac 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. associate-*r* 85.0%

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

      3. frac-times 85.4%

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

   8. Applied egg-rr 85.4%

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

      1. *-lft-identity 85.4%

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

      2. sub-neg 85.4%

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

      3. associate-*l/ 86.9%

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

      4. distribute-neg-frac2 86.9%

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

      5. *-commutative 86.9%

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

      6. associate-*l* 85.4%

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

      7. associate-/l* 85.4%

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

      8. *-commutative 85.4%

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

      9. *-commutative 85.4%

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

      10. distribute-rgt-neg-in 85.4%

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

      11. metadata-eval 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in h around 0 87.4%

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

       1. associate-*r/ 87.4%

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

       2. associate-*r* 87.4%

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

       3. *-commutative 87.4%

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

       4. *-commutative 87.4%

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

       5. *-commutative 87.4%

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

       6. associate-*l* 90.3%

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

       7. associate-*r* 90.3%

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

       8. *-commutative 90.3%

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

       9. *-commutative 90.3%

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

   13. Simplified 90.3%

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

   if 1.60000000000000013e-4 < M

   1. Initial program 77.2%

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

   3. Simplified 78.9%

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

      1. unpow2 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-*l/ 77.2%

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

      4. associate-*r/ 77.3%

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

      5. times-frac 77.2%

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

      6. clear-num 77.2%

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

      7. un-div-inv 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-*l/ 77.2%

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

      10. associate-*r/ 75.5%

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

      11. times-frac 77.2%

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

      12. associate-/l* 75.5%

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

      13. div-inv 75.5%

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

      14. associate-/r* 75.5%

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

      15. metadata-eval 75.5%

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

      16. times-frac 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. associate-*r* 75.5%

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

      3. frac-times 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. *-lft-identity 77.2%

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

      2. sub-neg 77.2%

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

      3. associate-*l/ 82.5%

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

      4. distribute-neg-frac2 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*l* 80.8%

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

      7. associate-/l* 80.8%

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

      8. *-commutative 80.8%

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

      9. *-commutative 80.8%

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

      10. distribute-rgt-neg-in 80.8%

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

      11. metadata-eval 80.8%

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

   10. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. inv-pow 80.8%

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

       3. *-commutative 80.8%

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

   12. Applied egg-rr 80.8%

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

       1. unpow-1 80.8%

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

       2. *-commutative 80.8%

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

       3. associate-/l* 82.6%

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

   14. Simplified 82.6%

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

   15. Taylor expanded in D around 0 80.8%

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

       1. associate-*r/ 80.8%

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

       2. *-commutative 80.8%

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

       3. associate-*l/ 80.8%

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

       4. associate-/r/ 80.8%

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

       5. *-commutative 80.8%

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

       6. associate-/l* 82.6%

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

       7. associate-/r* 82.7%

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

   17. Simplified 82.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 0.00016:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)}{\ell \cdot d}}{-2 \cdot \frac{d}{M \cdot D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{\frac{\frac{-2}{D}}{\frac{M}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(0.25 \cdot \frac{\frac{M\_m \cdot D\_m}{d}}{\frac{d}{M\_m \cdot D\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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
 (if (<= (/ h l) -1e-298)
   (*
    w0
    (sqrt
     (- 1.0 (* (/ h l) (* 0.25 (/ (/ (* M_m D_m) d) (/ d (* M_m D_m))))))))
   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 tmp;
	if ((h / l) <= -1e-298) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (0.25 * (((M_m * D_m) / d) / (d / (M_m * D_m)))))));
	} 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) :: tmp
    if ((h / l) <= (-1d-298)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (0.25d0 * (((m_m * d_m) / d) / (d / (m_m * d_m)))))))
    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 tmp;
	if ((h / l) <= -1e-298) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (0.25 * (((M_m * D_m) / d) / (d / (M_m * D_m)))))));
	} 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):
	tmp = 0
	if (h / l) <= -1e-298:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (0.25 * (((M_m * D_m) / d) / (d / (M_m * D_m)))))))
	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)
	tmp = 0.0
	if (Float64(h / l) <= -1e-298)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(0.25 * Float64(Float64(Float64(M_m * D_m) / d) / Float64(d / Float64(M_m * D_m))))))));
	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)
	tmp = 0.0;
	if ((h / l) <= -1e-298)
		tmp = w0 * sqrt((1.0 - ((h / l) * (0.25 * (((M_m * D_m) / d) / (d / (M_m * D_m)))))));
	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_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-298], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.99999999999999912e-299

   1. Initial program 81.0%

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

   3. Simplified 81.0%

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

      1. unpow2 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-*l/ 81.0%

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

      4. associate-*r/ 79.8%

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

      5. times-frac 81.0%

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

      6. clear-num 81.0%

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

      7. un-div-inv 81.0%

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

      8. *-commutative 81.0%

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

      9. associate-*l/ 81.0%

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

      10. associate-*r/ 79.8%

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

      11. times-frac 81.0%

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

      12. associate-/l* 79.8%

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

      13. div-inv 79.8%

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

      14. associate-/r* 79.8%

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

      15. metadata-eval 79.8%

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

      16. times-frac 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*r/ 86.9%

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

      2. associate-*r* 86.9%

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

      3. frac-times 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. *-un-lft-identity 88.1%

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

      2. associate-/l* 81.0%

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

      3. associate-/l* 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-/l* 79.1%

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

      6. associate-/r* 78.6%

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

   10. Applied egg-rr 78.6%

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

       1. *-lft-identity 78.6%

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

       2. *-commutative 78.6%

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

       3. associate-*r/ 79.8%

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

       4. associate-*r/ 79.8%

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

       5. associate-*l/ 79.8%

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

       6. *-commutative 79.8%

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

       7. *-commutative 79.8%

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

       8. times-frac 79.8%

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

       9. metadata-eval 79.8%

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

       10. associate-/l* 79.8%

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

       11. associate-/r* 81.0%

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

   12. Simplified 81.0%

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

       1. associate-*r/ 81.0%

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

       2. *-commutative 81.0%

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

   14. Applied egg-rr 81.0%

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

   if -9.99999999999999912e-299 < (/.f64 h l)

   1. Initial program 87.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in D around 0 97.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(0.25 \cdot \frac{\frac{M \cdot D}{d}}{\frac{d}{M \cdot D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(0.25 \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\frac{d}{M\_m \cdot D\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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
 (if (<= (/ h l) -1e-298)
   (*
    w0
    (sqrt
     (- 1.0 (* (/ h l) (* 0.25 (/ (* D_m (/ M_m d)) (/ d (* M_m D_m))))))))
   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 tmp;
	if ((h / l) <= -1e-298) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (0.25 * ((D_m * (M_m / d)) / (d / (M_m * D_m)))))));
	} 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) :: tmp
    if ((h / l) <= (-1d-298)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (0.25d0 * ((d_m * (m_m / d)) / (d / (m_m * d_m)))))))
    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 tmp;
	if ((h / l) <= -1e-298) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (0.25 * ((D_m * (M_m / d)) / (d / (M_m * D_m)))))));
	} 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):
	tmp = 0
	if (h / l) <= -1e-298:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (0.25 * ((D_m * (M_m / d)) / (d / (M_m * D_m)))))))
	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)
	tmp = 0.0
	if (Float64(h / l) <= -1e-298)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(0.25 * Float64(Float64(D_m * Float64(M_m / d)) / Float64(d / Float64(M_m * D_m))))))));
	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)
	tmp = 0.0;
	if ((h / l) <= -1e-298)
		tmp = w0 * sqrt((1.0 - ((h / l) * (0.25 * ((D_m * (M_m / d)) / (d / (M_m * D_m)))))));
	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_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-298], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.25 * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.99999999999999912e-299

   1. Initial program 81.0%

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

   3. Simplified 81.0%

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

      1. unpow2 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-*l/ 81.0%

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

      4. associate-*r/ 79.8%

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

      5. times-frac 81.0%

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

      6. clear-num 81.0%

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

      7. un-div-inv 81.0%

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

      8. *-commutative 81.0%

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

      9. associate-*l/ 81.0%

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

      10. associate-*r/ 79.8%

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

      11. times-frac 81.0%

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

      12. associate-/l* 79.8%

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

      13. div-inv 79.8%

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

      14. associate-/r* 79.8%

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

      15. metadata-eval 79.8%

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

      16. times-frac 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*r/ 86.9%

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

      2. associate-*r* 86.9%

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

      3. frac-times 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. *-un-lft-identity 88.1%

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

      2. associate-/l* 81.0%

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

      3. associate-/l* 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-/l* 79.1%

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

      6. associate-/r* 78.6%

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

   10. Applied egg-rr 78.6%

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

       1. *-lft-identity 78.6%

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

       2. *-commutative 78.6%

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

       3. associate-*r/ 79.8%

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

       4. associate-*r/ 79.8%

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

       5. associate-*l/ 79.8%

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

       6. *-commutative 79.8%

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

       7. *-commutative 79.8%

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

       8. times-frac 79.8%

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

       9. metadata-eval 79.8%

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

       10. associate-/l* 79.8%

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

       11. associate-/r* 81.0%

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

   12. Simplified 81.0%

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

   if -9.99999999999999912e-299 < (/.f64 h l)

   1. Initial program 87.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in D around 0 97.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(0.25 \cdot \frac{D \cdot \frac{M}{d}}{\frac{d}{M \cdot D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+161}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{w0}{{d}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e+161)
   w0
   (* -0.125 (* (* (* M_m D_m) (* M_m D_m)) (* (/ h l) (/ w0 (pow d 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e+161) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h / l) * (w0 / pow(d, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 2d+161) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * ((h / l) * (w0 / (d ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e+161) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h / l) * (w0 / Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 2e+161:
		tmp = w0
	else:
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h / l) * (w0 / math.pow(d, 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e+161)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(Float64(h / l) * Float64(w0 / (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 2e+161)
		tmp = w0;
	else
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((h / l) * (w0 / (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+161], w0, N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(w0 / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.0000000000000001e161

   1. Initial program 84.7%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in D around 0 78.1%

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

   if 2.0000000000000001e161 < (*.f64 M D)

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. unpow2 74.2%

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

      2. unpow2 74.2%

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

      3. *-commutative 74.2%

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

      4. associate-*l/ 74.2%

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

      5. associate-*r/ 74.2%

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

      6. times-frac 74.2%

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

      7. associate-*r/ 74.4%

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

      8. *-commutative 74.4%

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

      9. associate-/l* 74.4%

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

      10. div-inv 74.4%

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

      11. associate-/r* 74.4%

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

      12. metadata-eval 74.4%

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

   6. Applied egg-rr 74.4%

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

   7. Taylor expanded in h around 0 65.7%

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

      1. +-commutative 65.7%

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

      2. fma-define 65.7%

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

      3. associate-*r* 65.9%

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

      4. unpow2 65.9%

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

      5. unpow2 65.9%

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

      6. swap-sqr 65.9%

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

      7. unpow2 65.9%

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

      8. *-commutative 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in D around inf 65.7%

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

       1. associate-*r* 65.9%

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

       2. *-commutative 65.9%

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

       3. unpow2 65.9%

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

       4. unpow2 65.9%

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

       5. swap-sqr 65.9%

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

       6. unpow2 65.9%

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

       7. *-commutative 65.9%

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

       8. associate-*r/ 65.7%

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

       9. *-commutative 65.7%

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

       10. times-frac 73.9%

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

   12. Simplified 73.9%

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

       1. *-commutative 73.9%

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

       2. pow2 73.9%

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

   14. Applied egg-rr 73.9%

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

Alternative 10: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1650:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 1650.0)
   w0
   (* -0.125 (* (pow (* D_m (/ M_m d)) 2.0) (* h (/ w0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1650.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * (pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 1650.0d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d_m * (m_m / d)) ** 2.0d0) * (h * (w0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1650.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * (Math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 1650.0:
		tmp = w0
	else:
		tmp = -0.125 * (math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 1650.0)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(h * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 1650.0)
		tmp = w0;
	else
		tmp = -0.125 * (((D_m * (M_m / d)) ^ 2.0) * (h * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 1650.0], w0, N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1650

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in D around 0 76.5%

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

   if 1650 < M

   1. Initial program 76.8%

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

   3. Simplified 78.5%

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

      1. unpow2 78.5%

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

      2. unpow2 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-*l/ 76.8%

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

      5. associate-*r/ 76.8%

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

      6. times-frac 76.8%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. associate-/l* 82.4%

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

      10. div-inv 82.4%

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

      11. associate-/r* 82.4%

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

      12. metadata-eval 82.4%

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

   6. Applied egg-rr 82.4%

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

   7. Taylor expanded in h around 0 40.0%

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

      1. +-commutative 40.0%

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

      2. fma-define 40.0%

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

      3. associate-*r* 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

      6. swap-sqr 58.6%

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

      7. unpow2 58.6%

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

      8. *-commutative 58.6%

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

   9. Simplified 58.6%

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

   10. Taylor expanded in D around inf 27.6%

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

       1. associate-*r* 27.6%

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

       2. *-commutative 27.6%

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

       3. unpow2 27.6%

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

       4. unpow2 27.6%

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

       5. swap-sqr 30.3%

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

       6. unpow2 30.3%

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

       7. *-commutative 30.3%

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

       8. associate-*r/ 30.2%

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

       9. *-commutative 30.2%

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

       10. times-frac 31.9%

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

   12. Simplified 31.9%

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

   13. Taylor expanded in D around 0 27.6%

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

       1. associate-*r* 27.6%

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

       2. times-frac 29.3%

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

       3. *-commutative 29.3%

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

       4. associate-/l* 27.2%

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

       5. unpow2 27.2%

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

       6. unpow2 27.2%

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

       7. unpow2 27.2%

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

       8. times-frac 29.3%

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

       9. swap-sqr 32.4%

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

       10. associate-/l* 32.3%

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

       11. associate-/l* 32.4%

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

       12. unpow2 32.4%

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

       13. *-commutative 32.4%

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

       14. associate-/l* 32.5%

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

       15. associate-/l* 32.3%

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

   15. Simplified 32.3%

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

Alternative 11: 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 83.7%

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

3. Simplified 84.0%

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

5. Taylor expanded in D around 0 71.0%

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

Reproduce

herbie shell --seed 2024143 
(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))))))