Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 87.8%

Time: 15.4s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% 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: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\frac{D \cdot \left(D \cdot h\right)}{{\left(\frac{d}{M}\right)}^{2}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 2e+140)
   (* w0 (sqrt (- 1.0 (/ h (/ l (pow (/ (* D (* M 0.5)) d) 2.0))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (/ (/ (* D (* D h)) (pow (/ d M) 2.0)) l)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 2e+140) {
		tmp = w0 * sqrt((1.0 - (h / (l / pow(((D * (M * 0.5)) / d), 2.0)))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (D * h)) / pow((d / M), 2.0)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 2d+140) then
        tmp = w0 * sqrt((1.0d0 - (h / (l / (((d * (m * 0.5d0)) / d_1) ** 2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d * (d * h)) / ((d_1 / m) ** 2.0d0)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 2e+140) {
		tmp = w0 * Math.sqrt((1.0 - (h / (l / Math.pow(((D * (M * 0.5)) / d), 2.0)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * (D * h)) / Math.pow((d / M), 2.0)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 2e+140:
		tmp = w0 * math.sqrt((1.0 - (h / (l / math.pow(((D * (M * 0.5)) / d), 2.0)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * (D * h)) / math.pow((d / M), 2.0)) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 2e+140)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * Float64(D * h)) / (Float64(d / M) ^ 2.0)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 2e+140)
		tmp = w0 * sqrt((1.0 - (h / (l / (((D * (M * 0.5)) / d) ^ 2.0)))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (D * h)) / ((d / M) ^ 2.0)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e+140], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 2.00000000000000012e140

   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 83.2%

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

      1. associate-*r/ 88.6%

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

      2. frac-times 90.4%

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

      3. *-commutative 90.4%

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

      4. clear-num 90.3%

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

      5. *-commutative 90.3%

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

      6. div-inv 90.3%

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

      7. associate-*l* 91.2%

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

      8. associate-/r* 91.2%

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

      9. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. associate-/r/ 91.2%

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

      2. associate-*r/ 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*r/ 91.2%

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

      5. associate-*l* 91.2%

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

   7. Simplified 91.2%

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

      1. associate-*r/ 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-*l/ 90.4%

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

       2. *-un-lft-identity 90.4%

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

       3. associate-/l* 91.2%

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

   11. Applied egg-rr 91.2%

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

       1. associate-/l* 92.0%

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

       2. associate-/r/ 89.4%

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

       3. associate-*l/ 91.2%

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

       4. associate-*r* 91.2%

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

       5. *-commutative 91.2%

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

       6. *-commutative 91.2%

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

       7. associate-*l* 91.2%

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

       8. *-commutative 91.2%

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

   13. Simplified 91.2%

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

   if 2.00000000000000012e140 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

      1. associate-*r/ 55.7%

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

      2. frac-times 55.7%

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

      3. *-commutative 55.7%

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

      4. clear-num 55.7%

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

      5. *-commutative 55.7%

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

      6. div-inv 55.7%

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

      7. associate-*l* 55.7%

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

      8. associate-/r* 55.7%

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

      9. metadata-eval 55.7%

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

   5. Applied egg-rr 55.7%

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

      1. associate-/r/ 55.7%

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

      2. associate-*r/ 55.7%

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

      3. *-commutative 55.7%

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

      4. associate-*r/ 55.7%

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

      5. associate-*l* 55.7%

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

   7. Simplified 55.7%

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

      1. associate-*r/ 55.7%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in l around 0 51.8%

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

       1. *-commutative 51.8%

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

       2. *-commutative 51.8%

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

       3. associate-/r* 51.8%

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

       4. associate-*r/ 51.8%

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

       5. associate-*l/ 51.6%

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

       6. unpow2 51.6%

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

       7. associate-*r/ 54.9%

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

       8. unpow2 54.9%

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

       9. associate-*r/ 54.9%

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

       10. unpow2 54.9%

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

       11. associate-*r/ 54.9%

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

       12. associate-/l* 54.9%

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

   12. Simplified 62.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\frac{D \cdot \left(D \cdot h\right)}{{\left(\frac{d}{M}\right)}^{2}}}{\ell}}\\ \end{array} \]

Alternative 2: 87.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 2d-12) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in D around 0 71.5%

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

4. Final simplification 86.7%

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

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (/ (* D (* M (* M (/ h (* d d))))) l))))))
   (if (<= (/ h l) -1e-224)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     (* w0 (+ 1.0 (* (* D (/ (* D (* M (/ (* M (/ h d)) d))) l)) -0.125))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	} else if ((h / l) <= -1e-224) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	} else if ((h / l) <= -1e-224) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))))
	elif (h / l) <= -1e-224:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D * Float64(M * Float64(M * Float64(h / Float64(d * d))))) / l))))));
	elseif (Float64(h / l) <= -1e-224)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D * Float64(M * Float64(Float64(M * Float64(h / d)) / d))) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	elseif ((h / l) <= -1e-224)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D * N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-224], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 53.7%

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

   3. Simplified 49.4%

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

      1. associate-*r/ 79.5%

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

      2. frac-times 83.7%

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

      3. *-commutative 83.7%

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

      4. clear-num 83.5%

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

      5. *-commutative 83.5%

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

      6. div-inv 83.5%

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

      7. associate-*l* 83.6%

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

      8. associate-/r* 83.6%

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

      9. metadata-eval 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r/ 83.7%

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

      3. *-commutative 83.7%

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

      4. associate-*r/ 83.7%

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

      5. associate-*l* 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in l around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. times-frac 53.2%

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

      3. unpow2 53.2%

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

      4. associate-*l/ 53.3%

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

      5. unpow2 53.3%

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

      6. unpow2 53.3%

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

      7. times-frac 61.9%

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

      8. *-commutative 61.9%

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

      9. associate-*r* 66.2%

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

      10. *-commutative 66.2%

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

      11. *-commutative 66.2%

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

      12. times-frac 57.6%

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

      13. *-commutative 57.6%

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

      14. unpow2 57.6%

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

      15. associate-/l* 66.2%

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

      16. associate-/l/ 66.5%

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

      17. unpow2 66.5%

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

      18. associate-/r/ 66.7%

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

      19. *-commutative 66.7%

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

   10. Simplified 66.7%

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

       1. associate-*l/ 67.3%

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

   12. Applied egg-rr 75.5%

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

   if -inf.0 < (/.f64 h l) < -1e-224

   1. Initial program 83.9%

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

   3. Simplified 83.1%

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

   if -1e-224 < (/.f64 h l)

   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 82.0%

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

   4. Taylor expanded in D around 0 51.0%

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

      1. associate-*r/ 51.0%

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

      2. *-commutative 51.0%

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

      3. associate-*r/ 51.0%

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

      4. *-commutative 51.0%

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

      5. times-frac 58.0%

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

      6. unpow2 58.0%

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

      7. *-commutative 58.0%

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

      8. unpow2 58.0%

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

      9. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in D around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. times-frac 58.0%

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

      3. unpow2 58.0%

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

      4. associate-*l/ 64.0%

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

      5. unpow2 64.0%

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

      6. unpow2 64.0%

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

      7. times-frac 70.9%

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

      8. *-commutative 70.9%

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

      9. associate-*r* 78.8%

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

      10. *-commutative 78.8%

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

      11. *-commutative 78.8%

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

      12. times-frac 71.8%

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

      13. *-commutative 71.8%

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

      14. unpow2 71.8%

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

      15. associate-/l* 74.5%

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

      16. associate-/l/ 81.5%

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

      17. unpow2 81.5%

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

      18. associate-/r/ 81.5%

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

      19. *-commutative 81.5%

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

      20. unpow2 81.5%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in M around 0 78.0%

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

       1. unpow2 78.0%

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

       2. times-frac 84.9%

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

   12. Simplified 84.9%

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

       1. associate-*l/ 86.7%

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

       2. associate-*l/ 86.7%

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

   14. Applied egg-rr 86.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2}{\frac{M}{d}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e+308)
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (/ (* D (* M (* M (/ h (* d d))))) l))))))
   (if (<= (/ h l) -1e-224)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (/ 2.0 (/ M d))) 2.0)))))
     (* w0 (+ 1.0 (* (* D (/ (* D (* M (/ (* M (/ h d)) d))) l)) -0.125))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+308) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	} else if ((h / l) <= -1e-224) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / (2.0 / (M / d))), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-1d+308)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d * (m * (m * (h / (d_1 * d_1))))) / l)))))
    else if ((h / l) <= (-1d-224)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / (2.0d0 / (m / d_1))) ** 2.0d0))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d * (m * ((m * (h / d_1)) / d_1))) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+308) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	} else if ((h / l) <= -1e-224) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / (2.0 / (M / d))), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e+308:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))))
	elif (h / l) <= -1e-224:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / (2.0 / (M / d))), 2.0))))
	else:
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+308)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D * Float64(M * Float64(M * Float64(h / Float64(d * d))))) / l))))));
	elseif (Float64(h / l) <= -1e-224)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(2.0 / Float64(M / d))) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D * Float64(M * Float64(Float64(M * Float64(h / d)) / d))) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+308)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	elseif ((h / l) <= -1e-224)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / (2.0 / (M / d))) ^ 2.0))));
	else
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+308], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D * N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-224], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(2.0 / N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1e308

   1. Initial program 55.7%

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

   3. Simplified 51.5%

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

      1. associate-*r/ 80.4%

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

      2. frac-times 84.3%

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

      3. *-commutative 84.3%

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

      4. clear-num 84.2%

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

      5. *-commutative 84.2%

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

      6. div-inv 84.1%

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

      7. associate-*l* 84.3%

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

      8. associate-/r* 84.3%

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

      9. metadata-eval 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. associate-/r/ 84.3%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-*r/ 84.3%

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

      5. associate-*l* 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in l around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. times-frac 51.0%

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

      3. unpow2 51.0%

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

      4. associate-*l/ 51.2%

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

      5. unpow2 51.2%

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

      6. unpow2 51.2%

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

      7. times-frac 59.4%

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

      8. *-commutative 59.4%

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

      9. associate-*r* 67.6%

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

      10. *-commutative 67.6%

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

      11. *-commutative 67.6%

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

      12. times-frac 59.4%

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

      13. *-commutative 59.4%

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

      14. unpow2 59.4%

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

      15. associate-/l* 67.6%

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

      16. associate-/l/ 67.9%

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

      17. unpow2 67.9%

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

      18. associate-/r/ 68.1%

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

      19. *-commutative 68.1%

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

   10. Simplified 68.1%

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

       1. associate-*l/ 64.7%

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

   12. Applied egg-rr 76.5%

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

   if -1e308 < (/.f64 h l) < -1e-224

   1. Initial program 83.8%

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

   3. Simplified 82.9%

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

      1. associate-*l/ 82.9%

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

      2. associate-/l* 82.9%

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

   5. Applied egg-rr 82.9%

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

   if -1e-224 < (/.f64 h l)

   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 82.0%

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

   4. Taylor expanded in D around 0 51.0%

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

      1. associate-*r/ 51.0%

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

      2. *-commutative 51.0%

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

      3. associate-*r/ 51.0%

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

      4. *-commutative 51.0%

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

      5. times-frac 58.0%

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

      6. unpow2 58.0%

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

      7. *-commutative 58.0%

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

      8. unpow2 58.0%

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

      9. unpow2 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in D around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. times-frac 58.0%

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

      3. unpow2 58.0%

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

      4. associate-*l/ 64.0%

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

      5. unpow2 64.0%

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

      6. unpow2 64.0%

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

      7. times-frac 70.9%

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

      8. *-commutative 70.9%

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

      9. associate-*r* 78.8%

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

      10. *-commutative 78.8%

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

      11. *-commutative 78.8%

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

      12. times-frac 71.8%

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

      13. *-commutative 71.8%

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

      14. unpow2 71.8%

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

      15. associate-/l* 74.5%

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

      16. associate-/l/ 81.5%

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

      17. unpow2 81.5%

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

      18. associate-/r/ 81.5%

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

      19. *-commutative 81.5%

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

      20. unpow2 81.5%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in M around 0 78.0%

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

       1. unpow2 78.0%

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

       2. times-frac 84.9%

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

   12. Simplified 84.9%

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

       1. associate-*l/ 86.7%

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

       2. associate-*l/ 86.7%

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

   14. Applied egg-rr 86.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2}{\frac{M}{d}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{\frac{\ell}{D}} \cdot \left(0.25 \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e+130)
   (* w0 (sqrt (- 1.0 (/ h (/ l (pow (/ (* D (* M 0.5)) d) 2.0))))))
   (* w0 (sqrt (- 1.0 (* (/ D (/ l D)) (* 0.25 (/ h (* (/ d M) (/ d M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+130) {
		tmp = w0 * sqrt((1.0 - (h / (l / pow(((D * (M * 0.5)) / d), 2.0)))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 5d+130) then
        tmp = w0 * sqrt((1.0d0 - (h / (l / (((d * (m * 0.5d0)) / d_1) ** 2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / (l / d)) * (0.25d0 * (h / ((d_1 / m) * (d_1 / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+130) {
		tmp = w0 * Math.sqrt((1.0 - (h / (l / Math.pow(((D * (M * 0.5)) / d), 2.0)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 5e+130:
		tmp = w0 * math.sqrt((1.0 - (h / (l / math.pow(((D * (M * 0.5)) / d), 2.0)))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+130)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / Float64(l / D)) * Float64(0.25 * Float64(h / Float64(Float64(d / M) * Float64(d / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 5e+130)
		tmp = w0 * sqrt((1.0 - (h / (l / (((D * (M * 0.5)) / d) ^ 2.0)))));
	else
		tmp = w0 * sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+130], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 4.9999999999999996e130

   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 83.1%

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

      1. associate-*r/ 88.5%

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

      2. frac-times 90.3%

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

      3. *-commutative 90.3%

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

      4. clear-num 90.3%

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

      5. *-commutative 90.3%

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

      6. div-inv 90.3%

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

      7. associate-*l* 91.1%

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

      8. associate-/r* 91.1%

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

      9. metadata-eval 91.1%

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

   5. Applied egg-rr 91.1%

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

      1. associate-/r/ 91.1%

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

      2. associate-*r/ 91.1%

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

      3. *-commutative 91.1%

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

      4. associate-*r/ 91.1%

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

      5. associate-*l* 91.1%

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

   7. Simplified 91.1%

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

      1. associate-*r/ 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-*l/ 90.3%

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

       2. *-un-lft-identity 90.3%

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

       3. associate-/l* 91.1%

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

   11. Applied egg-rr 91.1%

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

       1. associate-/l* 92.0%

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

       2. associate-/r/ 89.4%

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

       3. associate-*l/ 91.1%

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

       4. associate-*r* 91.1%

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

       5. *-commutative 91.1%

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

       6. *-commutative 91.1%

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

       7. associate-*l* 91.1%

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

       8. *-commutative 91.1%

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

   13. Simplified 91.1%

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

   if 4.9999999999999996e130 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in D around 0 50.2%

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

      1. associate-*r/ 50.2%

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

      2. *-commutative 50.2%

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

      3. associate-*r/ 50.2%

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

      4. *-commutative 50.2%

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

      5. times-frac 50.0%

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

      6. associate-*l* 50.0%

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

      7. unpow2 50.0%

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

      8. associate-/l* 53.3%

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

      9. associate-/l* 53.3%

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

      10. unpow2 53.3%

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

      11. unpow2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in d around 0 53.3%

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

      1. unpow2 53.3%

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

      2. unpow2 53.3%

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

      3. times-frac 60.2%

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

   9. Simplified 60.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{\frac{\ell}{D}} \cdot \left(0.25 \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{M \cdot h}{d \cdot d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 7.2e-116)
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (/ h (pow (/ d M) 2.0)))))))
   (if (<= d 1.4e+119)
     (* w0 (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* M (/ (* M h) (* d d)))))))))
     (* w0 (+ 1.0 (* (* D (/ (* D (* M (/ (* M (/ h d)) d))) l)) -0.125))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.2e-116) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / pow((d / M), 2.0))))));
	} else if (d <= 1.4e+119) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (M * ((M * h) / (d * d))))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 7.2d-116) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * (h / ((d_1 / m) ** 2.0d0))))))
    else if (d_1 <= 1.4d+119) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / l) * (m * ((m * h) / (d_1 * d_1))))))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d * (m * ((m * (h / d_1)) / d_1))) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.2e-116) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / Math.pow((d / M), 2.0))))));
	} else if (d <= 1.4e+119) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (M * ((M * h) / (d * d))))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 7.2e-116:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / math.pow((d / M), 2.0))))))
	elif d <= 1.4e+119:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (M * ((M * h) / (d * d))))))))
	else:
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 7.2e-116)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(h / (Float64(d / M) ^ 2.0)))))));
	elseif (d <= 1.4e+119)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M * h) / Float64(d * d)))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D * Float64(M * Float64(Float64(M * Float64(h / d)) / d))) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 7.2e-116)
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / ((d / M) ^ 2.0))))));
	elseif (d <= 1.4e+119)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (M * ((M * h) / (d * d))))))));
	else
		tmp = w0 * (1.0 + ((D * ((D * (M * ((M * (h / d)) / d))) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 7.2e-116], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+119], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.19999999999999951e-116

   1. Initial program 81.3%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in D around 0 48.4%

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

      1. associate-*r/ 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*r/ 48.4%

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

      4. *-commutative 48.4%

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

      5. times-frac 52.7%

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

      6. unpow2 52.7%

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

      7. *-commutative 52.7%

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

      8. unpow2 52.7%

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

      9. unpow2 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in D around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. times-frac 52.7%

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

      3. unpow2 52.7%

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

      4. associate-*l/ 56.5%

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

      5. unpow2 56.5%

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

      6. unpow2 56.5%

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

      7. times-frac 62.2%

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

      8. *-commutative 62.2%

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

      9. associate-*r* 67.3%

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

      10. *-commutative 67.3%

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

      11. *-commutative 67.3%

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

      12. times-frac 61.6%

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

      13. *-commutative 61.6%

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

      14. unpow2 61.6%

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

      15. associate-/l* 62.2%

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

      16. associate-/l/ 67.5%

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

      17. unpow2 67.5%

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

      18. associate-/r/ 68.1%

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

      19. *-commutative 68.1%

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

      20. unpow2 68.1%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in D around 0 58.0%

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

       1. *-commutative 58.0%

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

       2. times-frac 61.6%

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

       3. associate-/l* 62.2%

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

       4. unpow2 62.2%

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

       5. unpow2 62.2%

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

       6. times-frac 72.4%

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

       7. unpow2 72.4%

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

   12. Simplified 72.4%

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

   if 7.19999999999999951e-116 < d < 1.40000000000000007e119

   1. Initial program 76.6%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 78.8%

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

      2. frac-times 83.2%

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

      3. *-commutative 83.2%

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

      4. clear-num 83.1%

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

      5. *-commutative 83.1%

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

      6. div-inv 83.1%

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

      7. associate-*l* 83.1%

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

      8. associate-/r* 83.1%

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

      9. metadata-eval 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. associate-/r/ 83.2%

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

      2. associate-*r/ 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-*r/ 83.2%

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

      5. associate-*l* 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in l around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. times-frac 61.2%

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

      3. unpow2 61.2%

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

      4. associate-*l/ 61.2%

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

      5. unpow2 61.2%

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

      6. unpow2 61.2%

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

      7. times-frac 61.1%

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

      8. *-commutative 61.1%

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

      9. associate-*r* 63.7%

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

      10. *-commutative 63.7%

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

      11. *-commutative 63.7%

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

      12. times-frac 63.7%

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

      13. *-commutative 63.7%

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

      14. unpow2 63.7%

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

      15. associate-/l* 65.9%

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

      16. associate-/l/ 65.9%

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

      17. unpow2 65.9%

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

      18. associate-/r/ 68.1%

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

      19. *-commutative 68.1%

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

   10. Simplified 65.7%

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

       1. associate-*r/ 67.9%

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

   12. Applied egg-rr 67.9%

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

   if 1.40000000000000007e119 < d

   1. Initial program 84.4%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in D around 0 52.3%

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

      1. associate-*r/ 52.3%

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

      2. *-commutative 52.3%

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

      3. associate-*r/ 52.3%

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

      4. *-commutative 52.3%

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

      5. times-frac 50.1%

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

      6. unpow2 50.1%

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

      7. *-commutative 50.1%

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

      8. unpow2 50.1%

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

      9. unpow2 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in D around 0 52.3%

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

      1. *-commutative 52.3%

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

      2. times-frac 50.1%

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

      3. unpow2 50.1%

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

      4. associate-*l/ 52.1%

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

      5. unpow2 52.1%

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

      6. unpow2 52.1%

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

      7. times-frac 58.6%

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

      8. *-commutative 58.6%

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

      9. associate-*r* 63.0%

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

      10. *-commutative 63.0%

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

      11. *-commutative 63.0%

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

      12. times-frac 56.4%

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

      13. *-commutative 56.4%

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

      14. unpow2 56.4%

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

      15. associate-/l* 62.4%

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

      16. associate-/l/ 77.0%

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

      17. unpow2 77.0%

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

      18. associate-/r/ 77.0%

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

      19. *-commutative 77.0%

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

      20. unpow2 77.0%

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

   9. Simplified 77.0%

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

   10. Taylor expanded in M around 0 72.9%

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

       1. unpow2 72.9%

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

       2. times-frac 77.2%

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

   12. Simplified 77.2%

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

       1. associate-*l/ 83.4%

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

       2. associate-*l/ 83.4%

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

   14. Applied egg-rr 83.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{M \cdot h}{d \cdot d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 7: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.5 \cdot 10^{-99}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 7.5e-99)
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (/ h (pow (/ d M) 2.0)))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (/ (* D (* M (* M (/ h (* d d))))) l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.5e-99) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / pow((d / M), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 7.5d-99) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * (h / ((d_1 / m) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d * (m * (m * (h / (d_1 * d_1))))) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.5e-99) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / Math.pow((d / M), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 7.5e-99:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / math.pow((d / M), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 7.5e-99)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(h / (Float64(d / M) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D * Float64(M * Float64(M * Float64(h / Float64(d * d))))) / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 7.5e-99)
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (h / ((d / M) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 7.5e-99], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D * N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.4999999999999999e-99

   1. Initial program 81.1%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in D around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-*r/ 48.1%

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

      4. *-commutative 48.1%

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

      5. times-frac 53.0%

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

      6. unpow2 53.0%

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

      7. *-commutative 53.0%

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

      8. unpow2 53.0%

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

      9. unpow2 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in D around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. times-frac 53.0%

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

      3. unpow2 53.0%

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

      4. associate-*l/ 56.7%

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

      5. unpow2 56.7%

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

      6. unpow2 56.7%

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

      7. times-frac 61.6%

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

      8. *-commutative 61.6%

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

      9. associate-*r* 66.7%

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

      10. *-commutative 66.7%

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

      11. *-commutative 66.7%

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

      12. times-frac 61.7%

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

      13. *-commutative 61.7%

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

      14. unpow2 61.7%

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

      15. associate-/l* 62.3%

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

      16. associate-/l/ 67.5%

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

      17. unpow2 67.5%

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

      18. associate-/r/ 68.1%

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

      19. *-commutative 68.1%

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

      20. unpow2 68.1%

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

   9. Simplified 66.3%

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

   10. Taylor expanded in D around 0 57.6%

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

       1. *-commutative 57.6%

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

       2. times-frac 61.7%

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

       3. associate-/l* 62.3%

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

       4. unpow2 62.3%

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

       5. unpow2 62.3%

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

       6. times-frac 72.3%

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

       7. unpow2 72.3%

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

   12. Simplified 72.3%

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

   if 7.4999999999999999e-99 < d

   1. Initial program 81.2%

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

   3. Simplified 79.1%

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

      1. associate-*r/ 84.5%

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

      2. frac-times 86.7%

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

      3. *-commutative 86.7%

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

      4. clear-num 86.7%

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

      5. *-commutative 86.7%

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

      6. div-inv 86.7%

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

      7. associate-*l* 87.7%

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

      8. associate-/r* 87.7%

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

      9. metadata-eval 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. associate-/r/ 87.7%

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

      2. associate-*r/ 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-*r/ 87.7%

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

      5. associate-*l* 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in l around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. times-frac 55.0%

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

      3. unpow2 55.0%

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

      4. associate-*l/ 56.1%

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

      5. unpow2 56.1%

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

      6. unpow2 56.1%

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

      7. times-frac 60.7%

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

      8. *-commutative 60.7%

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

      9. associate-*r* 64.2%

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

      10. *-commutative 64.2%

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

      11. *-commutative 64.2%

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

      12. times-frac 59.6%

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

      13. *-commutative 59.6%

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

      14. unpow2 59.6%

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

      15. associate-/l* 63.9%

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

      16. associate-/l/ 72.8%

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

      17. unpow2 72.8%

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

      18. associate-/r/ 73.9%

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

      19. *-commutative 73.9%

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

   10. Simplified 73.8%

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

       1. associate-*l/ 73.4%

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

   12. Applied egg-rr 78.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.5 \cdot 10^{-99}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{\frac{\ell}{D}} \cdot \left(0.25 \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.4e-90)
   (* w0 (sqrt (- 1.0 (* (/ D (/ l D)) (* 0.25 (/ h (* (/ d M) (/ d M))))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (/ (* D (* M (* M (/ h (* d d))))) l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.4e-90) {
		tmp = w0 * sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.4d-90) then
        tmp = w0 * sqrt((1.0d0 - ((d / (l / d)) * (0.25d0 * (h / ((d_1 / m) * (d_1 / m)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d * (m * (m * (h / (d_1 * d_1))))) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.4e-90) {
		tmp = w0 * Math.sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.4e-90:
		tmp = w0 * math.sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.4e-90)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / Float64(l / D)) * Float64(0.25 * Float64(h / Float64(Float64(d / M) * Float64(d / M))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D * Float64(M * Float64(M * Float64(h / Float64(d * d))))) / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.4e-90)
		tmp = w0 * sqrt((1.0 - ((D / (l / D)) * (0.25 * (h / ((d / M) * (d / M)))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * (M * (M * (h / (d * d))))) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.4e-90], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D * N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.3999999999999999e-90

   1. Initial program 80.4%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in D around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-*r/ 48.1%

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

      4. *-commutative 48.1%

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

      5. times-frac 52.9%

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

      6. associate-*l* 52.9%

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

      7. unpow2 52.9%

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

      8. associate-/l* 57.1%

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

      9. associate-/l* 57.1%

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

      10. unpow2 57.1%

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

      11. unpow2 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in d around 0 57.1%

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

      1. unpow2 57.1%

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

      2. unpow2 57.1%

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

      3. times-frac 68.5%

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

   9. Simplified 68.5%

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

   if 1.3999999999999999e-90 < d

   1. Initial program 82.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 87.2%

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

      2. frac-times 88.3%

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

      3. *-commutative 88.3%

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

      4. clear-num 88.3%

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

      5. *-commutative 88.3%

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

      6. div-inv 88.3%

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

      7. associate-*l* 89.3%

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

      8. associate-/r* 89.3%

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

      9. metadata-eval 89.3%

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

   5. Applied egg-rr 89.3%

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

      1. associate-/r/ 89.3%

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

      2. associate-*r/ 89.3%

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

      3. *-commutative 89.3%

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

      4. associate-*r/ 89.3%

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

      5. associate-*l* 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in l around 0 58.6%

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

      1. *-commutative 58.6%

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

      2. times-frac 56.3%

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

      3. unpow2 56.3%

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

      4. associate-*l/ 57.4%

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

      5. unpow2 57.4%

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

      6. unpow2 57.4%

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

      7. times-frac 62.2%

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

      8. *-commutative 62.2%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

      11. *-commutative 66.0%

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

      12. times-frac 61.1%

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

      13. *-commutative 61.1%

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

      14. unpow2 61.1%

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

      15. associate-/l* 65.6%

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

      16. associate-/l/ 74.9%

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

      17. unpow2 74.9%

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

      18. associate-/r/ 76.1%

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

      19. *-commutative 76.1%

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

   10. Simplified 76.0%

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

       1. associate-*l/ 75.5%

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

   12. Applied egg-rr 80.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{\frac{\ell}{D}} \cdot \left(0.25 \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right)}\\ \end{array} \]

Alternative 9: 77.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right) \cdot \frac{D}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.25e-87)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (* M (* M (/ h (* d d)))) (/ D l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-87) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((M * (M * (h / (d * d)))) * (D / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.25d-87) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((m * (m * (h / (d_1 * d_1)))) * (d / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-87) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((M * (M * (h / (d * d)))) * (D / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.25e-87:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((M * (M * (h / (d * d)))) * (D / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.25e-87)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(M * Float64(M * Float64(h / Float64(d * d)))) * Float64(D / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.25e-87)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((M * (M * (h / (d * d)))) * (D / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.25e-87], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000011e-87

   1. Initial program 82.7%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in D around 0 71.2%

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

   if 1.25000000000000011e-87 < M

   1. Initial program 77.6%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in D around 0 44.0%

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

      1. associate-*r/ 44.0%

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

      2. *-commutative 44.0%

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

      3. associate-*r/ 44.0%

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

      4. *-commutative 44.0%

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

      5. times-frac 46.5%

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

      6. unpow2 46.5%

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

      7. *-commutative 46.5%

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

      8. unpow2 46.5%

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

      9. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in D around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. times-frac 46.5%

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

      3. unpow2 46.5%

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

      4. associate-*l/ 50.2%

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

      5. unpow2 50.2%

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

      6. unpow2 50.2%

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

      7. times-frac 53.1%

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

      8. *-commutative 53.1%

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

      9. associate-*r* 57.0%

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

      10. *-commutative 57.0%

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

      11. *-commutative 57.0%

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

      12. times-frac 54.0%

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

      13. *-commutative 54.0%

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

      14. unpow2 54.0%

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

      15. associate-/l* 55.3%

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

      16. associate-/l/ 67.8%

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

      17. unpow2 67.8%

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

      18. associate-/r/ 70.3%

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

      19. *-commutative 70.3%

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

      20. unpow2 70.3%

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

   9. Simplified 69.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right) \cdot \frac{D}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 10: 77.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 8e+125)
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (* (/ M d) (/ h d))))))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8e+125) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 8d+125) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * (m * ((m / d_1) * (h / d_1)))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8e+125) {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 8e+125:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 8e+125)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M / d) * Float64(h / d))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 8e+125)
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M / d) * (h / d)))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 8e+125], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 7.9999999999999994e125

   1. Initial program 80.3%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in D around 0 49.8%

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

      1. associate-*r/ 49.8%

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

      2. *-commutative 49.8%

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

      3. associate-*r/ 49.8%

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

      4. *-commutative 49.8%

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

      5. times-frac 54.1%

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

      6. unpow2 54.1%

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

      7. *-commutative 54.1%

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

      8. unpow2 54.1%

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

      9. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in D around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. times-frac 54.1%

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

      3. unpow2 54.1%

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

      4. associate-*l/ 57.1%

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

      5. unpow2 57.1%

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

      6. unpow2 57.1%

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

      7. times-frac 61.1%

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

      8. *-commutative 61.1%

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

      9. associate-*r* 65.6%

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

      10. *-commutative 65.6%

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

      11. *-commutative 65.6%

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

      12. times-frac 61.6%

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

      13. *-commutative 61.6%

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

      14. unpow2 61.6%

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

      15. associate-/l* 62.2%

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

      16. associate-/l/ 66.3%

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

      17. unpow2 66.3%

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

      18. associate-/r/ 67.2%

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

      19. *-commutative 67.2%

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

      20. unpow2 67.2%

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

   9. Simplified 65.3%

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

   10. Taylor expanded in M around 0 64.7%

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

       1. unpow2 64.7%

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

       2. times-frac 70.1%

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

   12. Simplified 70.1%

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

   if 7.9999999999999994e125 < d

   1. Initial program 84.4%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in D around 0 83.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 78.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{-141}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.5e-141)
   w0
   (* w0 (+ 1.0 (* (* D (/ (* D (* M (* M (/ h (* d d))))) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.5e-141) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * (M * (h / (d * d))))) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.5d-141) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((d * ((d * (m * (m * (h / (d_1 * d_1))))) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.5e-141) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * ((D * (M * (M * (h / (d * d))))) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.5e-141:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((D * ((D * (M * (M * (h / (d * d))))) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.5e-141)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D * Float64(M * Float64(M * Float64(h / Float64(d * d))))) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.5e-141)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((D * ((D * (M * (M * (h / (d * d))))) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.5e-141], w0, N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D * N[(M * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.5000000000000003e-141

   1. Initial program 81.7%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in D around 0 70.0%

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

   if 3.5000000000000003e-141 < M

   1. Initial program 80.1%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in D around 0 48.0%

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

      1. associate-*r/ 48.0%

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

      2. *-commutative 48.0%

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

      3. associate-*r/ 48.0%

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

      4. *-commutative 48.0%

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

      5. times-frac 50.2%

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

      6. unpow2 50.2%

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

      7. *-commutative 50.2%

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

      8. unpow2 50.2%

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

      9. unpow2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in D around 0 48.0%

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

      1. *-commutative 48.0%

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

      2. times-frac 50.2%

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

      3. unpow2 50.2%

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

      4. associate-*l/ 53.5%

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

      5. unpow2 53.5%

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

      6. unpow2 53.5%

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

      7. times-frac 57.3%

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

      8. *-commutative 57.3%

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

      9. associate-*r* 60.7%

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

      10. *-commutative 60.7%

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

      11. *-commutative 60.7%

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

      12. times-frac 56.9%

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

      13. *-commutative 56.9%

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

      14. unpow2 56.9%

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

      15. associate-/l* 58.0%

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

      16. associate-/l/ 69.2%

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

      17. unpow2 69.2%

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

      18. associate-/r/ 71.4%

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

      19. *-commutative 71.4%

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

      20. unpow2 71.4%

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

   9. Simplified 70.3%

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

       1. associate-*l/ 71.7%

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

   11. Applied egg-rr 71.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{-141}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 12: 72.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e+51)
   w0
   (* -0.125 (* (/ (* D D) (* d d)) (/ (* w0 (* h (* M M))) l)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+51) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 9d+51) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * ((w0 * (h * (m * m))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+51) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e+51:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (h * (M * M))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e+51)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e+51)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (h * (M * M))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e+51], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999999e51

   1. Initial program 82.9%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in D around 0 70.8%

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

   if 8.9999999999999999e51 < M

   1. Initial program 73.7%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in D around 0 34.1%

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

      1. associate-*r/ 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*r/ 34.1%

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

      4. *-commutative 34.1%

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

      5. times-frac 34.1%

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

      6. unpow2 34.1%

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

      7. *-commutative 34.1%

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

      8. unpow2 34.1%

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

      9. unpow2 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in D around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. times-frac 34.1%

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

      3. unpow2 34.1%

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

      4. associate-*l/ 36.1%

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

      5. unpow2 36.1%

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

      6. unpow2 36.1%

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

      7. times-frac 38.8%

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

      8. *-commutative 38.8%

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

      9. associate-*r* 42.9%

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

      10. *-commutative 42.9%

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

      11. *-commutative 42.9%

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

      12. times-frac 40.2%

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

      13. *-commutative 40.2%

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

      14. unpow2 40.2%

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

      15. associate-/l* 42.2%

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

      16. associate-/l/ 62.1%

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

      17. unpow2 62.1%

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

      18. associate-/r/ 66.0%

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

      19. *-commutative 66.0%

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

      20. unpow2 66.0%

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

   9. Simplified 64.0%

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

   10. Taylor expanded in D around 0 42.2%

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

       1. *-commutative 42.2%

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

       2. times-frac 40.2%

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

       3. associate-/l* 42.2%

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

       4. unpow2 42.2%

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

       5. unpow2 42.2%

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

       6. times-frac 62.4%

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

       7. unpow2 62.4%

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

   12. Simplified 62.4%

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

   13. Taylor expanded in D around inf 21.0%

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

       1. associate-/l* 21.0%

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

       2. *-commutative 21.0%

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

       3. associate-/l* 21.0%

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

       4. times-frac 20.9%

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

       5. unpow2 20.9%

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

       6. unpow2 20.9%

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

       7. unpow2 20.9%

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

   15. Simplified 20.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 13: 79.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((d * ((d * (m * ((m * (h / d_1)) / d_1))) / l)) * (-0.125d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

3. Simplified 79.6%

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

4. Taylor expanded in D around 0 50.3%

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

   1. associate-*r/ 50.3%

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

   2. *-commutative 50.3%

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

   3. associate-*r/ 50.3%

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

   4. *-commutative 50.3%

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

   5. times-frac 53.3%

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

   6. unpow2 53.3%

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

   7. *-commutative 53.3%

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

   8. unpow2 53.3%

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

   9. unpow2 53.3%

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

6. Simplified 53.3%

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

7. Taylor expanded in D around 0 50.3%

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

   1. *-commutative 50.3%

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

   2. times-frac 53.3%

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

   3. unpow2 53.3%

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

   4. associate-*l/ 56.1%

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

   5. unpow2 56.1%

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

   6. unpow2 56.1%

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

   7. times-frac 60.6%

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

   8. *-commutative 60.6%

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

   9. associate-*r* 65.1%

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

   10. *-commutative 65.1%

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

   11. *-commutative 65.1%

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

   12. times-frac 60.6%

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

   13. *-commutative 60.6%

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

   14. unpow2 60.6%

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

   15. associate-/l* 62.2%

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

   16. associate-/l/ 68.4%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{\color{blue}{\frac{\frac{{d}^{2}}{M}}{M}}}\right)\right) \cdot -0.125\right) \]

   17. unpow2 68.4%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h}{\frac{\frac{\color{blue}{d \cdot d}}{M}}{M}}\right)\right) \cdot -0.125\right) \]

   18. associate-/r/ 69.1%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \color{blue}{\left(\frac{h}{\frac{d \cdot d}{M}} \cdot M\right)}\right)\right) \cdot -0.125\right) \]

   19. *-commutative 69.1%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \color{blue}{\left(M \cdot \frac{h}{\frac{d \cdot d}{M}}\right)}\right)\right) \cdot -0.125\right) \]

   20. unpow2 69.1%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{h}{\frac{\color{blue}{{d}^{2}}}{M}}\right)\right)\right) \cdot -0.125\right) \]

9. Simplified 67.6%

   \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)\right)\right)} \cdot -0.125\right) \]

10. Taylor expanded in M around 0 66.3%

    \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \cdot -0.125\right) \]
11. Step-by-step derivation

    1. unpow2 66.3%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{M \cdot h}{\color{blue}{d \cdot d}}\right)\right)\right) \cdot -0.125\right) \]

    2. times-frac 71.5%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right) \cdot -0.125\right) \]

12. Simplified 71.5%

    \[\leadsto w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right) \cdot -0.125\right) \]
13. Step-by-step derivation

    1. associate-*l/ 75.3%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \color{blue}{\frac{D \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{h}{d}\right)\right)}{\ell}}\right) \cdot -0.125\right) \]

    2. associate-*l/ 75.3%

       \[\leadsto w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \color{blue}{\frac{M \cdot \frac{h}{d}}{d}}\right)}{\ell}\right) \cdot -0.125\right) \]

14. Applied egg-rr 75.3%

    \[\leadsto w0 \cdot \left(1 + \left(D \cdot \color{blue}{\frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}}\right) \cdot -0.125\right) \]

15. Final simplification 75.3%

    \[\leadsto w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)}{\ell}\right) \cdot -0.125\right) \]

Alternative 14: 70.1% accurate, 216.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.1%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 79.6%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

4. Taylor expanded in D around 0 67.6%

   \[\leadsto \color{blue}{w0} \]

5. Final simplification 67.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023271 
(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))))))