Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 90.1%

Time: 17.8s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.5\right)\right) \cdot \frac{t\_0}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d)))))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e+154)
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_0)))))
     (* w0 (sqrt (- 1.0 (* h (* (* (/ M_m d) (* D_m 0.5)) (/ 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)) <= -4e+154) {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * (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)) <= (-4d+154)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m / d) * (d_m * 0.5d0)) * (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)) <= -4e+154) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * (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)) <= -4e+154:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * (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)) <= -4e+154)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m / d) * Float64(D_m * 0.5)) * Float64(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)) <= -4e+154)
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	else
		tmp = w0 * sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * (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], -4e+154], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.00000000000000015e154

   1. Initial program 58.0%

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

   3. Simplified 61.5%

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

      1. associate-*r/ 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*l* 62.9%

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

      4. *-commutative 62.9%

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

      5. associate-/r* 62.9%

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

      6. associate-*l/ 59.2%

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

      7. associate-*r/ 61.6%

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

      8. *-commutative 61.6%

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

      9. associate-/r* 61.6%

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

      10. associate-*l/ 59.2%

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

      11. associate-*r/ 62.9%

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

      12. associate-*l* 59.1%

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

      13. unpow2 59.1%

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

      14. *-commutative 59.1%

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

   6. Applied egg-rr 59.2%

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

      1. associate-*r/ 57.9%

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

      2. frac-times 59.2%

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

      3. associate-*l/ 61.5%

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

      4. unpow2 61.5%

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

      5. associate-*r* 68.8%

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

      6. frac-times 62.8%

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

      7. associate-*r/ 68.8%

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

      8. *-un-lft-identity 68.8%

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

      9. times-frac 68.8%

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

      10. metadata-eval 68.8%

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

      11. frac-times 62.8%

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

      12. associate-*r/ 68.8%

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

      13. *-un-lft-identity 68.8%

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

      14. times-frac 68.8%

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

      15. metadata-eval 68.8%

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

   8. Applied egg-rr 68.8%

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

   if -4.00000000000000015e154 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. unpow1 87.1%

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

      2. associate-*l/ 97.1%

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

      3. associate-/l* 97.0%

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

      4. associate-*r/ 96.8%

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

      5. times-frac 97.0%

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

      6. associate-*r/ 96.8%

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

   8. Simplified 96.8%

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

      1. associate-*r/ 97.0%

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

      2. unpow2 97.0%

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

      3. *-un-lft-identity 97.0%

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

      4. times-frac 97.8%

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

      5. frac-times 96.8%

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

      6. associate-*r/ 97.8%

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

      7. *-un-lft-identity 97.8%

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

      8. times-frac 97.8%

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

      9. metadata-eval 97.8%

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

      10. frac-times 96.8%

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

      11. associate-*r/ 97.8%

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

      12. *-un-lft-identity 97.8%

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

      13. times-frac 97.8%

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

      14. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

       1. /-rgt-identity 97.8%

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

       2. associate-*r* 97.8%

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

   12. Applied egg-rr 97.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-170}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{D\_m \cdot \left(M\_m \cdot \left(D\_m \cdot \frac{\frac{M\_m}{d}}{d \cdot 4}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(D\_m \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{M\_m}{d}\right)}{\ell \cdot \left(d \cdot 4\right)}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2e-170)
   (*
    w0
    (sqrt (- 1.0 (* h (/ (* D_m (* M_m (* D_m (/ (/ M_m d) (* d 4.0))))) l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* h (* D_m (/ (* D_m (* M_m (/ M_m d))) (* l (* d 4.0))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-170) {
		tmp = w0 * sqrt((1.0 - (h * ((D_m * (M_m * (D_m * ((M_m / d) / (d * 4.0))))) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2d-170) then
        tmp = w0 * sqrt((1.0d0 - (h * ((d_m * (m_m * (d_m * ((m_m / d) / (d * 4.0d0))))) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (d_m * ((d_m * (m_m * (m_m / d))) / (l * (d * 4.0d0)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-170) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((D_m * (M_m * (D_m * ((M_m / d) / (d * 4.0))))) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2e-170:
		tmp = w0 * math.sqrt((1.0 - (h * ((D_m * (M_m * (D_m * ((M_m / d) / (d * 4.0))))) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2e-170)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(D_m * Float64(M_m * Float64(D_m * Float64(Float64(M_m / d) / Float64(d * 4.0))))) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(D_m * Float64(Float64(D_m * Float64(M_m * Float64(M_m / d))) / Float64(l * Float64(d * 4.0))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2e-170)
		tmp = w0 * sqrt((1.0 - (h * ((D_m * (M_m * (D_m * ((M_m / d) / (d * 4.0))))) / l))));
	else
		tmp = w0 * sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2e-170], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(D$95$m * N[(M$95$m * N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(D$95$m * N[(N[(D$95$m * N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.99999999999999997e-170

   1. Initial program 79.2%

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

   3. Simplified 79.8%

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

      1. associate-*r/ 89.0%

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

      2. unpow2 89.0%

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

      3. associate-*l* 89.8%

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

      4. *-commutative 89.8%

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

      5. associate-/r* 89.8%

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

      6. associate-*l/ 87.5%

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

      7. associate-*r/ 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-/r* 88.6%

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

      10. associate-*l/ 87.5%

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

      11. associate-*r/ 89.8%

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

      12. associate-*l* 89.0%

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

      13. unpow2 89.0%

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

      14. *-commutative 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. unpow2 89.0%

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

      2. *-commutative 89.0%

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

      3. associate-/r/ 88.4%

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

      4. associate-*r/ 88.1%

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

      5. frac-times 88.4%

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

      6. associate-*l/ 88.4%

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

      7. clear-num 88.4%

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

      8. frac-times 88.4%

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

      9. *-un-lft-identity 88.4%

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

      10. clear-num 88.4%

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

      11. associate-/r/ 89.0%

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

      12. *-commutative 89.0%

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

      13. associate-*r/ 87.5%

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

      14. frac-times 89.0%

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

      15. associate-*r/ 87.5%

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

      16. clear-num 87.5%

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

      17. *-un-lft-identity 87.5%

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

      18. times-frac 89.0%

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

      19. clear-num 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

      2. *-rgt-identity 89.0%

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

      3. times-frac 89.0%

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

      4. times-frac 87.5%

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

      5. associate-/r/ 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*r/ 87.5%

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

      8. *-rgt-identity 87.5%

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

      9. times-frac 86.1%

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

      10. *-commutative 86.1%

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

      11. *-commutative 86.1%

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

      12. associate-/l* 85.7%

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

      13. *-commutative 85.7%

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

      14. associate-*r* 85.7%

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

      15. metadata-eval 85.7%

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

   10. Simplified 85.7%

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

       1. associate-/l* 85.7%

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

       2. associate-*r/ 85.5%

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

       3. *-commutative 85.5%

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

   12. Applied egg-rr 85.5%

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

       1. associate-*r/ 86.1%

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

       2. associate-/l* 85.7%

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

   14. Applied egg-rr 85.7%

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

       1. associate-*r/ 86.1%

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

       2. associate-*r* 85.8%

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

       3. *-commutative 85.8%

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

       4. associate-*l/ 87.2%

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

       5. *-commutative 87.2%

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

       6. associate-/r/ 87.2%

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

       7. associate-/l* 88.4%

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

       8. associate-*r* 87.9%

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

       9. associate-*r/ 87.2%

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

       10. associate-*r* 87.8%

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

       11. associate-/l* 86.6%

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

       12. associate-/r/ 86.0%

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

       13. *-commutative 86.0%

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

   16. Simplified 82.4%

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

   if 1.99999999999999997e-170 < M

   1. Initial program 76.9%

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

   3. Simplified 78.9%

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

      1. associate-*r/ 80.1%

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

      2. unpow2 80.1%

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

      3. associate-*l* 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-/r* 82.1%

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

      6. associate-*l/ 81.1%

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

      7. associate-*r/ 82.1%

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

      8. *-commutative 82.1%

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

      9. associate-/r* 82.1%

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

      10. associate-*l/ 82.0%

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

      11. associate-*r/ 83.0%

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

      12. associate-*l* 80.1%

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

      13. unpow2 80.1%

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

      14. *-commutative 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. unpow2 80.1%

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

      2. *-commutative 80.1%

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

      3. associate-/r/ 80.1%

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

      4. associate-*r/ 79.1%

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

      5. frac-times 80.1%

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

      6. associate-*l/ 80.1%

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

      7. clear-num 80.1%

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

      8. frac-times 80.1%

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

      9. *-un-lft-identity 80.1%

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

      10. clear-num 80.1%

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

      11. associate-/r/ 80.1%

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

      12. *-commutative 80.1%

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

      13. associate-*r/ 79.1%

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

      14. frac-times 80.1%

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

      15. associate-*r/ 79.1%

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

      16. clear-num 79.1%

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

      17. *-un-lft-identity 79.1%

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

      18. times-frac 80.1%

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

      19. clear-num 80.1%

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

   8. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. times-frac 80.1%

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

      4. times-frac 79.1%

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

      5. associate-/r/ 79.1%

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

      6. *-commutative 79.1%

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

      7. associate-*r/ 79.1%

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

      8. *-rgt-identity 79.1%

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

      9. times-frac 76.1%

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

      10. *-commutative 76.1%

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

      11. *-commutative 76.1%

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

      12. associate-/l* 79.1%

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

      13. *-commutative 79.1%

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

      14. associate-*r* 79.1%

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

      15. metadata-eval 79.1%

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

   10. Simplified 79.1%

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

       1. associate-/l* 78.1%

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

       2. associate-*r/ 76.1%

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

       3. *-commutative 76.1%

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

   12. Applied egg-rr 76.1%

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

       1. *-un-lft-identity 76.1%

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

       2. associate-/l/ 76.9%

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

       3. associate-*l* 73.9%

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

   14. Applied egg-rr 73.9%

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

       1. *-lft-identity 73.9%

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

       2. associate-/l* 73.9%

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

       3. *-commutative 73.9%

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

       4. associate-*l* 70.8%

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

   16. Simplified 70.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-170}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{D \cdot \left(M \cdot \left(D \cdot \frac{\frac{M}{d}}{d \cdot 4}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(D \cdot \frac{D \cdot \left(M \cdot \frac{M}{d}\right)}{\ell \cdot \left(d \cdot 4\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.8× speedup

Math

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

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
  (sqrt
   (-
    1.0
    (* h (* (* (/ M_m d) (* D_m 0.5)) (/ (* D_m (* 0.5 (/ M_m d))) 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 * sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * ((D_m * (0.5 * (M_m / d))) / 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 * sqrt((1.0d0 - (h * (((m_m / d) * (d_m * 0.5d0)) * ((d_m * (0.5d0 * (m_m / d))) / 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 * Math.sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * ((D_m * (0.5 * (M_m / d))) / 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 * math.sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * ((D_m * (0.5 * (M_m / d))) / 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 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m / d) * Float64(D_m * 0.5)) * Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) / 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 * sqrt((1.0 - (h * (((M_m / d) * (D_m * 0.5)) * ((D_m * (0.5 * (M_m / d))) / 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[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 79.4%

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

6. Applied egg-rr 79.4%

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

   1. unpow1 79.4%

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

   2. associate-*l/ 85.7%

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

   3. associate-/l* 84.5%

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

   4. associate-*r/ 84.4%

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

   5. times-frac 84.5%

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

   6. associate-*r/ 84.4%

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

8. Simplified 84.4%

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

   1. associate-*r/ 84.5%

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

   2. unpow2 84.5%

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

   3. *-un-lft-identity 84.5%

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

   4. times-frac 86.9%

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

   5. frac-times 85.1%

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

   6. associate-*r/ 86.9%

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

   7. *-un-lft-identity 86.9%

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

   8. times-frac 86.9%

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

   9. metadata-eval 86.9%

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

   10. frac-times 85.1%

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

   11. associate-*r/ 86.9%

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

   12. *-un-lft-identity 86.9%

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

   13. times-frac 86.9%

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

   14. metadata-eval 86.9%

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

10. Applied egg-rr 86.9%

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

    1. /-rgt-identity 86.9%

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

    2. associate-*r* 86.9%

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

12. Applied egg-rr 86.9%

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

13. Final simplification 86.9%

    \[\leadsto w0 \cdot \sqrt{1 - h \cdot \left(\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\ell}\right)} \]
14. Add Preprocessing

Alternative 4: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ w0 \cdot \sqrt{1 - \frac{h \cdot \frac{t\_0 \cdot t\_0}{4}}{\ell}} \end{array} \end{array} \]

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 (/ M_m d))))
   (* w0 (sqrt (- 1.0 (/ (* h (/ (* t_0 t_0) 4.0)) 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 * (M_m / d);
	return w0 * sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / 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
    real(8) :: t_0
    t_0 = d_m * (m_m / d)
    code = w0 * sqrt((1.0d0 - ((h * ((t_0 * t_0) / 4.0d0)) / 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) {
	double t_0 = D_m * (M_m / d);
	return w0 * Math.sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / 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):
	t_0 = D_m * (M_m / d)
	return w0 * math.sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / 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)
	t_0 = Float64(D_m * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(t_0 * t_0) / 4.0)) / 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)
	t_0 = D_m * (M_m / d);
	tmp = w0 * sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / 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_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.3%

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

3. Simplified 79.4%

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

   1. associate-*r/ 85.7%

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

   2. unpow2 85.7%

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

   3. associate-*l* 86.9%

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

   4. *-commutative 86.9%

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

   5. associate-/r* 86.9%

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

   6. associate-*l/ 85.1%

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

   7. associate-*r/ 86.2%

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

   8. *-commutative 86.2%

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

   9. associate-/r* 86.2%

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

   10. associate-*l/ 85.5%

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

   11. associate-*r/ 87.3%

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

   12. associate-*l* 85.6%

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

   13. unpow2 85.6%

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

   14. *-commutative 85.6%

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

6. Applied egg-rr 85.7%

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

   1. associate-*r/ 85.1%

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

   2. frac-times 85.7%

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

   3. unpow2 85.7%

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

   4. associate-*l/ 85.7%

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

   5. associate-*l/ 85.7%

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

   6. frac-times 85.7%

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

   7. metadata-eval 85.7%

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

8. Applied egg-rr 85.7%

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

Alternative 5: 86.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(D\_m \cdot \frac{M\_m}{d}\right) \cdot \frac{M\_m}{4 \cdot \frac{d}{D\_m}}\right)}{\ell}} \end{array} \]

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
  (sqrt (- 1.0 (/ (* h (* (* D_m (/ M_m d)) (/ M_m (* 4.0 (/ d D_m))))) 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 * sqrt((1.0 - ((h * ((D_m * (M_m / d)) * (M_m / (4.0 * (d / D_m))))) / 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 * sqrt((1.0d0 - ((h * ((d_m * (m_m / d)) * (m_m / (4.0d0 * (d / d_m))))) / 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 * Math.sqrt((1.0 - ((h * ((D_m * (M_m / d)) * (M_m / (4.0 * (d / D_m))))) / 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 * math.sqrt((1.0 - ((h * ((D_m * (M_m / d)) * (M_m / (4.0 * (d / D_m))))) / 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 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(D_m * Float64(M_m / d)) * Float64(M_m / Float64(4.0 * Float64(d / D_m))))) / 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 * sqrt((1.0 - ((h * ((D_m * (M_m / d)) * (M_m / (4.0 * (d / D_m))))) / 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[Sqrt[N[(1.0 - N[(N[(h * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(4.0 * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 79.4%

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

   1. associate-*r/ 85.7%

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

   2. unpow2 85.7%

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

   3. associate-*l* 86.9%

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

   4. *-commutative 86.9%

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

   5. associate-/r* 86.9%

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

   6. associate-*l/ 85.1%

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

   7. associate-*r/ 86.2%

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

   8. *-commutative 86.2%

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

   9. associate-/r* 86.2%

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

   10. associate-*l/ 85.5%

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

   11. associate-*r/ 87.3%

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

   12. associate-*l* 85.6%

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

   13. unpow2 85.6%

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

   14. *-commutative 85.6%

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

6. Applied egg-rr 85.7%

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

   1. unpow2 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-/r/ 85.3%

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

   4. associate-*r/ 84.8%

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

   5. frac-times 85.3%

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

   6. associate-*l/ 85.3%

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

   7. frac-times 84.1%

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

   8. associate-/l* 84.1%

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

8. Applied egg-rr 84.1%

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

   1. *-commutative 84.1%

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

   2. *-commutative 84.1%

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

   3. associate-/l* 85.3%

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

   4. associate-*r* 85.3%

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

   5. metadata-eval 85.3%

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

10. Simplified 85.3%

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

Alternative 6: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{D\_m \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\frac{d \cdot 4}{D\_m}}\right)}{\ell}} \end{array} \]

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
  (sqrt (- 1.0 (* h (/ (* D_m (* (/ M_m d) (/ M_m (/ (* d 4.0) D_m)))) 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 * sqrt((1.0 - (h * ((D_m * ((M_m / d) * (M_m / ((d * 4.0) / D_m)))) / 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 * sqrt((1.0d0 - (h * ((d_m * ((m_m / d) * (m_m / ((d * 4.0d0) / d_m)))) / 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 * Math.sqrt((1.0 - (h * ((D_m * ((M_m / d) * (M_m / ((d * 4.0) / D_m)))) / 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 * math.sqrt((1.0 - (h * ((D_m * ((M_m / d) * (M_m / ((d * 4.0) / D_m)))) / 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 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(D_m * Float64(Float64(M_m / d) * Float64(M_m / Float64(Float64(d * 4.0) / D_m)))) / 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 * sqrt((1.0 - (h * ((D_m * ((M_m / d) * (M_m / ((d * 4.0) / D_m)))) / 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[Sqrt[N[(1.0 - N[(h * N[(N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / N[(N[(d * 4.0), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 79.4%

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

   1. associate-*r/ 85.7%

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

   2. unpow2 85.7%

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

   3. associate-*l* 86.9%

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

   4. *-commutative 86.9%

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

   5. associate-/r* 86.9%

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

   6. associate-*l/ 85.1%

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

   7. associate-*r/ 86.2%

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

   8. *-commutative 86.2%

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

   9. associate-/r* 86.2%

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

   10. associate-*l/ 85.5%

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

   11. associate-*r/ 87.3%

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

   12. associate-*l* 85.6%

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

   13. unpow2 85.6%

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

   14. *-commutative 85.6%

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

6. Applied egg-rr 85.7%

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

   1. unpow2 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-/r/ 85.3%

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

   4. associate-*r/ 84.8%

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

   5. frac-times 85.3%

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

   6. associate-*l/ 85.3%

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

   7. frac-times 84.1%

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

   8. associate-/l* 84.1%

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

8. Applied egg-rr 84.1%

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

   1. *-commutative 84.1%

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

   2. *-commutative 84.1%

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

   3. associate-/l* 85.3%

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

   4. associate-*r* 85.3%

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

   5. metadata-eval 85.3%

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

10. Simplified 85.3%

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

    1. associate-/l* 84.5%

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

    2. associate-*l* 83.0%

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

    3. associate-*r/ 83.0%

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

    4. *-commutative 83.0%

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

12. Applied egg-rr 83.0%

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

Alternative 7: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \left(D\_m \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{M\_m}{d}\right)}{\ell \cdot \left(d \cdot 4\right)}\right)} \end{array} \]

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
  (sqrt (- 1.0 (* h (* D_m (/ (* D_m (* M_m (/ M_m d))) (* l (* d 4.0)))))))))

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) {
	return w0 * sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
}

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 * sqrt((1.0d0 - (h * (d_m * ((d_m * (m_m * (m_m / d))) / (l * (d * 4.0d0)))))))
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 * Math.sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
}

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 * math.sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))))

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 * sqrt(Float64(1.0 - Float64(h * Float64(D_m * Float64(Float64(D_m * Float64(M_m * Float64(M_m / d))) / Float64(l * Float64(d * 4.0))))))))
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 * sqrt((1.0 - (h * (D_m * ((D_m * (M_m * (M_m / d))) / (l * (d * 4.0)))))));
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[Sqrt[N[(1.0 - N[(h * N[(D$95$m * N[(N[(D$95$m * N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 79.4%

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

   1. associate-*r/ 85.7%

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

   2. unpow2 85.7%

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

   3. associate-*l* 86.9%

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

   4. *-commutative 86.9%

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

   5. associate-/r* 86.9%

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

   6. associate-*l/ 85.1%

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

   7. associate-*r/ 86.2%

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

   8. *-commutative 86.2%

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

   9. associate-/r* 86.2%

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

   10. associate-*l/ 85.5%

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

   11. associate-*r/ 87.3%

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

   12. associate-*l* 85.6%

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

   13. unpow2 85.6%

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

   14. *-commutative 85.6%

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

6. Applied egg-rr 85.7%

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

   1. unpow2 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-/r/ 85.3%

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

   4. associate-*r/ 84.8%

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

   5. frac-times 85.3%

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

   6. associate-*l/ 85.3%

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

   7. clear-num 85.3%

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

   8. frac-times 85.3%

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

   9. *-un-lft-identity 85.3%

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

   10. clear-num 85.3%

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

   11. associate-/r/ 85.7%

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

   12. *-commutative 85.7%

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

   13. associate-*r/ 84.4%

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

   14. frac-times 85.7%

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

   15. associate-*r/ 84.4%

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

   16. clear-num 84.4%

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

   17. *-un-lft-identity 84.4%

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

   18. times-frac 85.6%

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

   19. clear-num 85.6%

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

8. Applied egg-rr 85.6%

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

   1. *-commutative 85.6%

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

   2. *-rgt-identity 85.6%

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

   3. times-frac 85.6%

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

   4. times-frac 84.4%

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

   5. associate-/r/ 84.4%

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

   6. *-commutative 84.4%

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

   7. associate-*r/ 84.4%

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

   8. *-rgt-identity 84.4%

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

   9. times-frac 82.3%

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

   10. *-commutative 82.3%

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

   11. *-commutative 82.3%

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

   12. associate-/l* 83.3%

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

   13. *-commutative 83.3%

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

   14. associate-*r* 83.3%

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

   15. metadata-eval 83.3%

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

10. Simplified 83.3%

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

    1. associate-/l* 82.9%

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

    2. associate-*r/ 82.0%

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

    3. *-commutative 82.0%

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

12. Applied egg-rr 82.0%

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

    1. *-un-lft-identity 82.0%

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

    2. associate-/l/ 80.9%

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

    3. associate-*l* 79.4%

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

14. Applied egg-rr 79.4%

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

    1. *-lft-identity 79.4%

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

    2. associate-/l* 80.3%

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

    3. *-commutative 80.3%

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

    4. associate-*l* 77.2%

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

16. Simplified 77.2%

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

17. Final simplification 77.2%

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

Alternative 8: 68.2% 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 78.3%

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

3. Simplified 79.4%

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

5. Taylor expanded in D around 0 63.5%

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

Reproduce

herbie shell --seed 2024167 
(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))))))