Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.7%

Time: 17.2s

Alternatives: 6

Speedup: 1.6×

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: 81.0% 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: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+87}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_0\right) \cdot \frac{D\_m \cdot 0.5}{\frac{d}{M\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot \left(h \cdot t\_0\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d)))))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+87)
     (* w0 (sqrt (- 1.0 (* (* (/ h l) t_0) (/ (* D_m 0.5) (/ d M_m))))))
     (* w0 (sqrt (- 1.0 (/ (* t_0 (* h t_0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+87) {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * ((D_m * 0.5) / (d / M_m)))));
	} else {
		tmp = w0 * sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m_m / d))
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-5d+87)) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * t_0) * ((d_m * 0.5d0) / (d / m_m)))))
    else
        tmp = w0 * sqrt((1.0d0 - ((t_0 * (h * t_0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+87) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_0) * ((D_m * 0.5) / (d / M_m)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (0.5 * (M_m / d))
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+87:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * t_0) * ((D_m * 0.5) / (d / M_m)))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * (h * t_0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+87)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_0) * Float64(Float64(D_m * 0.5) / Float64(d / M_m))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(h * t_0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (0.5 * (M_m / d));
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+87)
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * ((D_m * 0.5) / (d / M_m)))));
	else
		tmp = w0 * sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+87], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.9999999999999998e87

   1. Initial program 65.9%

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

   3. Simplified 67.3%

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

      1. unpow2 67.3%

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

      2. associate-*l* 74.1%

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

      3. frac-times 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-*l/ 74.1%

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

      6. associate-/r/ 70.1%

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

      7. frac-times 67.4%

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

      8. *-commutative 67.4%

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

      9. associate-*l/ 70.1%

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

      10. associate-/r/ 71.4%

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

      11. associate-*l* 63.3%

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

      12. unpow2 63.3%

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

      13. associate-*r/ 59.3%

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

   6. Applied egg-rr 59.3%

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

      1. associate-*l/ 63.3%

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

      2. unpow2 63.3%

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

      3. associate-*r* 71.4%

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

      4. div-inv 71.4%

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

      5. metadata-eval 71.4%

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

      6. div-inv 71.4%

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

      7. metadata-eval 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*l/ 67.3%

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

      3. associate-*l/ 67.4%

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

      4. frac-times 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*r* 61.8%

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

      7. associate-*l/ 64.4%

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

      8. associate-*r/ 63.1%

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

      9. *-commutative 63.1%

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

      10. *-un-lft-identity 63.1%

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

      11. times-frac 63.1%

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

      12. metadata-eval 63.1%

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

   10. Applied egg-rr 63.1%

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

       1. *-commutative 63.1%

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

       2. *-commutative 63.1%

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

       3. times-frac 68.7%

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

       4. associate-*l/ 70.0%

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

       5. associate-/l* 74.1%

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

       6. *-commutative 74.1%

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

   12. Simplified 74.1%

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

   if -4.9999999999999998e87 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 88.2%

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

   3. Simplified 87.9%

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

      1. unpow2 87.9%

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

      2. associate-*l* 88.6%

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

      3. frac-times 88.0%

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

      4. *-commutative 88.0%

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

      5. associate-*l/ 88.6%

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

      6. associate-/r/ 88.6%

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

      7. frac-times 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-*l/ 88.6%

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

      10. associate-/r/ 89.4%

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

      11. associate-*l* 88.7%

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

      12. unpow2 88.7%

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

      13. associate-*r/ 97.0%

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

   6. Applied egg-rr 97.0%

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

      1. add-sqr-sqrt 81.4%

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

      2. pow2 81.4%

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

      3. *-commutative 81.4%

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

      4. sqrt-prod 52.2%

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

      5. unpow2 52.2%

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

      6. sqrt-prod 34.9%

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

      7. add-sqr-sqrt 52.7%

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

      8. div-inv 52.7%

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

      9. metadata-eval 52.7%

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

   8. Applied egg-rr 52.7%

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

      1. associate-*l/ 53.2%

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

      2. *-commutative 53.2%

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

      3. associate-*l/ 53.0%

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

      4. *-commutative 53.0%

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

   10. Simplified 53.0%

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

       1. unpow2 53.0%

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

       2. associate-*r/ 53.0%

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

       3. associate-*l/ 52.4%

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

       4. *-commutative 52.4%

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

       5. associate-*r/ 52.7%

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

       6. associate-*l/ 52.7%

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

       7. *-commutative 52.7%

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

       8. swap-sqr 52.2%

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

       9. add-sqr-sqrt 97.0%

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

       10. associate-*r* 98.4%

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

   12. Applied egg-rr 98.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+87}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{D \cdot 0.5}{\frac{d}{M}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty \lor \neg \left(\frac{h}{\ell} \leq -1 \cdot 10^{-310}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)\right) \cdot \frac{D\_m \cdot 0.5}{\frac{d}{M\_m}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (or (<= (/ h l) (- INFINITY)) (not (<= (/ h l) -1e-310)))
   (* w0 (+ 1.0 (* -0.125 (/ (* h (pow (* M_m (/ D_m d)) 2.0)) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ h l) (* D_m (* 0.5 (/ M_m d)))) (/ (* D_m 0.5) (/ d M_m))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) <= -((double) INFINITY)) || !((h / l) <= -1e-310)) {
		tmp = w0 * (1.0 + (-0.125 * ((h * pow((M_m * (D_m / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) <= -Double.POSITIVE_INFINITY) || !((h / l) <= -1e-310)) {
		tmp = w0 * (1.0 + (-0.125 * ((h * Math.pow((M_m * (D_m / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) <= -math.inf) or not ((h / l) <= -1e-310):
		tmp = w0 * (1.0 + (-0.125 * ((h * math.pow((M_m * (D_m / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if ((Float64(h / l) <= Float64(-Inf)) || !(Float64(h / l) <= -1e-310))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * (Float64(M_m * Float64(D_m / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(D_m * Float64(0.5 * Float64(M_m / d)))) * Float64(Float64(D_m * 0.5) / Float64(d / M_m))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) <= -Inf) || ~(((h / l) <= -1e-310)))
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M_m * (D_m / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[Or[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(h / l), $MachinePrecision], -1e-310]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -inf.0 or -9.999999999999969e-311 < (/.f64 h l)

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in D around 0 60.1%

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

      1. associate-*r* 64.0%

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

      2. add-sqr-sqrt 29.7%

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

      3. times-frac 28.9%

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

      4. pow-prod-down 35.1%

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

      5. sqrt-prod 35.1%

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

      6. unpow2 35.1%

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

      7. sqrt-prod 17.1%

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

      8. add-sqr-sqrt 34.3%

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

      9. sqrt-prod 35.1%

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

      10. unpow2 35.1%

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

      11. sqrt-prod 20.2%

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

      12. add-sqr-sqrt 39.8%

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

   7. Applied egg-rr 39.8%

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

   8. Taylor expanded in D around 0 60.1%

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

      1. associate-*r* 64.0%

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

      2. unpow2 64.0%

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

      3. unpow2 64.0%

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

      4. swap-sqr 77.2%

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

      5. unpow2 77.2%

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

      6. *-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. times-frac 69.1%

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

      9. unpow2 69.1%

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

      10. associate-/r* 80.0%

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

      11. unpow2 80.0%

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

      12. associate-*r/ 80.7%

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

      13. associate-*l/ 80.7%

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

      14. associate-*l/ 81.5%

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

      15. associate-*l/ 82.3%

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

      16. unpow2 82.3%

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

      17. *-commutative 82.3%

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

   10. Simplified 82.3%

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

       1. associate-*l/ 94.3%

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

   12. Applied egg-rr 94.3%

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

   if -inf.0 < (/.f64 h l) < -9.999999999999969e-311

   1. Initial program 82.6%

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

   3. Simplified 83.0%

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

      1. unpow2 83.0%

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

      2. associate-*l* 86.8%

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

      3. frac-times 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*l/ 86.8%

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

      6. associate-/r/ 84.6%

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

      7. frac-times 83.4%

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

      8. *-commutative 83.4%

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

      9. associate-*l/ 84.6%

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

      10. associate-/r/ 85.7%

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

      11. associate-*l* 81.2%

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

      12. unpow2 81.2%

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

      13. associate-*r/ 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. associate-*l/ 81.2%

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

      2. unpow2 81.2%

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

      3. associate-*r* 85.7%

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

      4. div-inv 85.7%

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

      5. metadata-eval 85.7%

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

      6. div-inv 85.7%

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

      7. metadata-eval 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 83.4%

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

      3. associate-*l/ 83.4%

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

      4. frac-times 78.0%

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

      5. *-commutative 78.0%

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

      6. associate-*r* 78.0%

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

      7. associate-*l/ 79.5%

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

      8. associate-*r/ 78.4%

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

      9. *-commutative 78.4%

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

      10. *-un-lft-identity 78.4%

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

      11. times-frac 78.4%

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

      12. metadata-eval 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. *-commutative 78.4%

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

       2. *-commutative 78.4%

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

       3. times-frac 83.8%

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

       4. associate-*l/ 84.5%

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

       5. associate-/l* 86.8%

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

       6. *-commutative 86.8%

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

   12. Simplified 86.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty \lor \neg \left(\frac{h}{\ell} \leq -1 \cdot 10^{-310}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{D \cdot 0.5}{\frac{d}{M}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+209}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot 0.5\right)\right)}{d \cdot \ell} \cdot \left(\frac{D\_m}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)\right) \cdot \frac{D\_m \cdot 0.5}{\frac{d}{M\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -1e+209)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ (* h (* M_m (* D_m 0.5))) (* d l)) (* (/ D_m d) (* M_m 0.5))))))
   (if (<= (/ h l) -1e-310)
     (*
      w0
      (sqrt
       (-
        1.0
        (* (* (/ h l) (* D_m (* 0.5 (/ M_m d)))) (/ (* D_m 0.5) (/ d M_m))))))
     (* w0 (+ 1.0 (* -0.125 (/ (* h (pow (* M_m (/ D_m d)) 2.0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+209) {
		tmp = w0 * sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / (d * l)) * ((D_m / d) * (M_m * 0.5)))));
	} else if ((h / l) <= -1e-310) {
		tmp = w0 * sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h * pow((M_m * (D_m / d)), 2.0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-1d+209)) then
        tmp = w0 * sqrt((1.0d0 - (((h * (m_m * (d_m * 0.5d0))) / (d * l)) * ((d_m / d) * (m_m * 0.5d0)))))
    else if ((h / l) <= (-1d-310)) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * (d_m * (0.5d0 * (m_m / d)))) * ((d_m * 0.5d0) / (d / m_m)))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h * ((m_m * (d_m / d)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+209) {
		tmp = w0 * Math.sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / (d * l)) * ((D_m / d) * (M_m * 0.5)))));
	} else if ((h / l) <= -1e-310) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h * Math.pow((M_m * (D_m / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -1e+209:
		tmp = w0 * math.sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / (d * l)) * ((D_m / d) * (M_m * 0.5)))))
	elif (h / l) <= -1e-310:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((h * math.pow((M_m * (D_m / d)), 2.0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+209)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(M_m * Float64(D_m * 0.5))) / Float64(d * l)) * Float64(Float64(D_m / d) * Float64(M_m * 0.5))))));
	elseif (Float64(h / l) <= -1e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(D_m * Float64(0.5 * Float64(M_m / d)))) * Float64(Float64(D_m * 0.5) / Float64(d / M_m))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * (Float64(M_m * Float64(D_m / d)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+209)
		tmp = w0 * sqrt((1.0 - (((h * (M_m * (D_m * 0.5))) / (d * l)) * ((D_m / d) * (M_m * 0.5)))));
	elseif ((h / l) <= -1e-310)
		tmp = w0 * sqrt((1.0 - (((h / l) * (D_m * (0.5 * (M_m / d)))) * ((D_m * 0.5) / (d / M_m)))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M_m * (D_m / d)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+209], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1.0000000000000001e209

   1. Initial program 61.2%

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

   3. Simplified 62.6%

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

      1. unpow2 62.6%

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

      2. associate-*l* 63.4%

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

      3. frac-times 60.7%

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

      4. *-commutative 60.7%

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

      5. associate-*l/ 63.4%

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

      6. associate-/r/ 60.9%

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

      7. frac-times 59.6%

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

      8. *-commutative 59.6%

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

      9. associate-*l/ 60.9%

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

      10. associate-/r/ 62.2%

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

      11. associate-*l* 61.4%

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

      12. unpow2 61.4%

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

      13. associate-*r/ 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-*l/ 61.4%

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

      2. unpow2 61.4%

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

      3. associate-*r* 62.2%

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

      4. div-inv 62.2%

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

      5. metadata-eval 62.2%

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

      6. div-inv 62.2%

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

      7. metadata-eval 62.2%

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

   8. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l/ 59.6%

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

      3. frac-times 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*r* 82.4%

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

   10. Applied egg-rr 82.4%

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

   if -1.0000000000000001e209 < (/.f64 h l) < -9.999999999999969e-311

   1. Initial program 82.1%

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

   3. Simplified 83.0%

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

      1. unpow2 83.0%

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

      2. associate-*l* 87.4%

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

      3. frac-times 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-*l/ 87.4%

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

      6. associate-/r/ 85.7%

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

      7. frac-times 83.9%

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

      8. *-commutative 83.9%

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

      9. associate-*l/ 85.7%

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

      10. associate-/r/ 86.5%

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

      11. associate-*l* 81.3%

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

      12. unpow2 81.3%

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

      13. associate-*r/ 78.6%

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

   6. Applied egg-rr 78.6%

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

      1. associate-*l/ 81.3%

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

      2. unpow2 81.3%

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

      3. associate-*r* 86.5%

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

      4. div-inv 86.5%

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

      5. metadata-eval 86.5%

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

      6. div-inv 86.5%

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

      7. metadata-eval 86.5%

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

   8. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*l/ 83.9%

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

      3. associate-*l/ 83.9%

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

      4. frac-times 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*r* 77.5%

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

      7. associate-*l/ 78.3%

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

      8. associate-*r/ 77.5%

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

      9. *-commutative 77.5%

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

      10. *-un-lft-identity 77.5%

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

      11. times-frac 77.5%

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

      12. metadata-eval 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. *-commutative 77.5%

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

       2. *-commutative 77.5%

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

       3. times-frac 83.9%

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

       4. associate-*l/ 84.7%

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

       5. associate-/l* 87.5%

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

       6. *-commutative 87.5%

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

   12. Simplified 87.5%

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

   if -9.999999999999969e-311 < (/.f64 h l)

   1. Initial program 89.2%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in D around 0 59.5%

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

      1. associate-*r* 64.1%

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

      2. add-sqr-sqrt 30.7%

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

      3. times-frac 30.7%

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

      4. pow-prod-down 37.0%

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

      5. sqrt-prod 37.0%

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

      6. unpow2 37.0%

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

      7. sqrt-prod 18.0%

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

      8. add-sqr-sqrt 36.9%

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

      9. sqrt-prod 37.8%

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

      10. unpow2 37.8%

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

      11. sqrt-prod 21.6%

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

      12. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

   8. Taylor expanded in D around 0 59.5%

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

      1. associate-*r* 64.1%

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

      2. unpow2 64.1%

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

      3. unpow2 64.1%

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

      4. swap-sqr 78.5%

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

      5. unpow2 78.5%

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

      6. *-commutative 78.5%

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

      7. *-commutative 78.5%

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

      8. times-frac 74.8%

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

      9. unpow2 74.8%

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

      10. associate-/r* 87.4%

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

      11. unpow2 87.4%

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

      12. associate-*r/ 88.3%

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

      13. associate-*l/ 88.3%

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

      14. associate-*l/ 89.2%

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

      15. associate-*l/ 89.2%

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

      16. unpow2 89.2%

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

      17. *-commutative 89.2%

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

   10. Simplified 89.2%

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

       1. associate-*l/ 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+209}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{d \cdot \ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{D \cdot 0.5}{\frac{d}{M}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (/ (* h (pow (* M_m (/ D_m d)) 2.0)) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((h * pow((M_m * (D_m / d)), 2.0)) / l)));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * (1.0d0 + ((-0.125d0) * ((h * ((m_m * (d_m / d)) ** 2.0d0)) / l)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((h * Math.pow((M_m * (D_m / d)), 2.0)) / l)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * (1.0 + (-0.125 * ((h * math.pow((M_m * (D_m / d)), 2.0)) / l)))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * (Float64(M_m * Float64(D_m / d)) ^ 2.0)) / l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((h * ((M_m * (D_m / d)) ^ 2.0)) / l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Simplified 82.3%

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

5. Taylor expanded in D around 0 53.3%

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

   1. associate-*r* 56.0%

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

   2. add-sqr-sqrt 28.0%

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

   3. times-frac 27.6%

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

   4. pow-prod-down 32.8%

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

   5. sqrt-prod 32.8%

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

   6. unpow2 32.8%

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

   7. sqrt-prod 17.6%

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

   8. add-sqr-sqrt 30.2%

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

   9. sqrt-prod 30.6%

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

   10. unpow2 30.6%

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

   11. sqrt-prod 20.0%

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

   12. add-sqr-sqrt 35.9%

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

7. Applied egg-rr 35.9%

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

8. Taylor expanded in D around 0 53.3%

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

   1. associate-*r* 56.0%

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

   2. unpow2 56.0%

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

   3. unpow2 56.0%

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

   4. swap-sqr 67.3%

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

   5. unpow2 67.3%

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

   6. *-commutative 67.3%

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

   7. *-commutative 67.3%

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

   8. times-frac 63.7%

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

   9. unpow2 63.7%

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

   10. associate-/r* 73.4%

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

   11. unpow2 73.4%

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

   12. associate-*r/ 75.0%

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

   13. associate-*l/ 74.9%

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

   14. associate-*l/ 75.3%

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

   15. associate-*l/ 75.8%

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

   16. unpow2 75.8%

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

   17. *-commutative 75.8%

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

10. Simplified 75.8%

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

    1. associate-*l/ 81.8%

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

12. Applied egg-rr 81.8%

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

13. Final simplification 81.8%

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

Alternative 5: 78.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m}{d}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (/ D_m d))))
   (if (<= (* M_m D_m) 1e-13)
     w0
     (* w0 (+ 1.0 (* -0.125 (* (/ h l) (* t_0 t_0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (D_m / d);
	double tmp;
	if ((M_m * D_m) <= 1e-13) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_m / d)
    if ((m_m * d_m) <= 1d-13) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / l) * (t_0 * t_0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (D_m / d);
	double tmp;
	if ((M_m * D_m) <= 1e-13) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = M_m * (D_m / d)
	tmp = 0
	if (M_m * D_m) <= 1e-13:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(D_m / d))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-13)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = M_m * (D_m / d);
	tmp = 0.0;
	if ((M_m * D_m) <= 1e-13)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-13], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-13

   1. Initial program 84.2%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in D around 0 75.0%

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

   if 1e-13 < (*.f64 M D)

   1. Initial program 72.5%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in D around 0 41.2%

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

      1. associate-*r* 45.6%

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

      2. add-sqr-sqrt 25.8%

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

      3. times-frac 27.9%

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

      4. pow-prod-down 34.5%

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

      5. sqrt-prod 34.5%

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

      6. unpow2 34.5%

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

      7. sqrt-prod 19.0%

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

      8. add-sqr-sqrt 27.7%

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

      9. sqrt-prod 27.7%

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

      10. unpow2 27.7%

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

      11. sqrt-prod 21.2%

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

      12. add-sqr-sqrt 36.8%

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

   7. Applied egg-rr 36.8%

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

   8. Taylor expanded in D around 0 41.2%

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

      1. associate-*r* 45.6%

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

      2. unpow2 45.6%

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

      3. unpow2 45.6%

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

      4. swap-sqr 58.7%

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

      5. unpow2 58.7%

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

      6. *-commutative 58.7%

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

      7. *-commutative 58.7%

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

      8. times-frac 56.5%

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

      9. unpow2 56.5%

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

      10. associate-/r* 58.9%

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

      11. unpow2 58.9%

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

      12. associate-*r/ 63.2%

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

      13. associate-*l/ 63.2%

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

      14. associate-*l/ 65.3%

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

      15. associate-*l/ 65.5%

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

      16. unpow2 65.5%

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

      17. *-commutative 65.5%

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

   10. Simplified 65.5%

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

       1. unpow2 65.5%

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

   12. Applied egg-rr 65.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 82.3%

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

5. Taylor expanded in D around 0 68.2%

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

6. Final simplification 68.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))))