Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 90.1%

Time: 21.0s

Alternatives: 4

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: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m / d) * (D_m / 2.0);
	return w0 * sqrt((1.0 - ((t_0 / l) * (t_0 * h))));
}

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 = (m_m / d) * (d_m / 2.0d0)
    code = w0 * sqrt((1.0d0 - ((t_0 / l) * (t_0 * h))))
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (M_m / d) * (D_m / 2.0)
	return w0 * math.sqrt((1.0 - ((t_0 / l) * (t_0 * h))))

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(Float64(M_m / d) * Float64(D_m / 2.0))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(t_0 * h)))))
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 = (M_m / d) * (D_m / 2.0);
	tmp = w0 * sqrt((1.0 - ((t_0 / l) * (t_0 * h))));
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[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.4%

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

3. Simplified 79.0%

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

   1. clear-num 79.0%

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

   2. un-div-inv 80.0%

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

   3. clear-num 80.0%

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

   4. un-div-inv 80.4%

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

   5. associate-/r/ 80.4%

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

6. Applied egg-rr 80.4%

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

   1. associate-*l/ 80.4%

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

   2. *-commutative 80.4%

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

   3. associate-*l/ 80.4%

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

   4. associate-/l/ 80.5%

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

   5. associate-/r/ 80.5%

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

8. Simplified 80.5%

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

   1. unpow2 80.5%

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

   2. div-inv 80.5%

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

   3. times-frac 90.3%

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

   4. associate-*l/ 90.3%

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

   5. associate-*l/ 90.3%

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

10. Applied egg-rr 90.3%

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

    1. associate-*l/ 89.5%

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

    2. associate-/l* 88.1%

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

    3. associate-*l/ 88.1%

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

    4. *-commutative 88.1%

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

    5. associate-/r/ 88.1%

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

    6. /-rgt-identity 88.1%

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

    7. associate-*l/ 88.8%

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

    8. associate-/l* 89.6%

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

    9. associate-*l/ 89.6%

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

    10. *-commutative 89.6%

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

12. Simplified 89.6%

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

13. Final simplification 89.6%

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

Alternative 2: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-183}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M\_m}{d} \cdot \frac{D\_m}{2}}{\ell} \cdot \left(0.5 \cdot \frac{D\_m \cdot \left(M\_m \cdot h\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D\_m}{d} \cdot \frac{\frac{M\_m}{2}}{\ell}\right) \cdot \left(h \cdot \left(M\_m \cdot \frac{D\_m \cdot 0.5}{d}\right)\right)}\\ \end{array} \end{array} \]

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 (<= d 1.3e-183)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ (* (/ M_m d) (/ D_m 2.0)) l) (* 0.5 (/ (* D_m (* M_m h)) d))))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ D_m d) (/ (/ M_m 2.0) l)) (* h (* M_m (/ (* D_m 0.5) d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.3e-183) {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) * (D_m / 2.0)) / l) * (0.5 * ((D_m * (M_m * h)) / d)))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 1.3d-183) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m / d) * (d_m / 2.0d0)) / l) * (0.5d0 * ((d_m * (m_m * h)) / d)))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m / d) * ((m_m / 2.0d0) / l)) * (h * (m_m * ((d_m * 0.5d0) / d))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.3e-183) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) * (D_m / 2.0)) / l) * (0.5 * ((D_m * (M_m * h)) / d)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 1.3e-183:
		tmp = w0 * math.sqrt((1.0 - ((((M_m / d) * (D_m / 2.0)) / l) * (0.5 * ((D_m * (M_m * h)) / d)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 1.3e-183)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(D_m / 2.0)) / l) * Float64(0.5 * Float64(Float64(D_m * Float64(M_m * h)) / d))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(Float64(M_m / 2.0) / l)) * Float64(h * Float64(M_m * Float64(Float64(D_m * 0.5) / d)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 1.3e-183)
		tmp = w0 * sqrt((1.0 - ((((M_m / d) * (D_m / 2.0)) / l) * (0.5 * ((D_m * (M_m * h)) / d)))));
	else
		tmp = w0 * sqrt((1.0 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 1.3e-183], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(N[(D$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m / 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.2999999999999999e-183

   1. Initial program 78.2%

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

   3. Simplified 77.6%

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

      1. clear-num 77.5%

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

      2. un-div-inv 78.6%

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

      3. clear-num 78.6%

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

      4. un-div-inv 79.2%

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

      5. associate-/r/ 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-commutative 79.2%

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

      3. associate-*l/ 79.2%

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

      4. associate-/l/ 79.5%

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

      5. associate-/r/ 79.5%

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

   8. Simplified 79.5%

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

      1. unpow2 79.5%

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

      2. div-inv 79.5%

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

      3. times-frac 88.8%

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

      4. associate-*l/ 88.8%

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

      5. associate-*l/ 88.8%

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

   10. Applied egg-rr 88.8%

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

       1. associate-*l/ 87.5%

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

       2. associate-/l* 85.7%

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

       3. associate-*l/ 85.7%

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

       4. *-commutative 85.7%

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

       5. associate-/r/ 85.7%

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

       6. /-rgt-identity 85.7%

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

       7. associate-*l/ 86.8%

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

       8. associate-/l* 88.1%

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

       9. associate-*l/ 88.1%

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

       10. *-commutative 88.1%

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

   12. Simplified 88.1%

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

   13. Taylor expanded in M around 0 82.6%

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

   if 1.2999999999999999e-183 < d

   1. Initial program 81.1%

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

   3. Simplified 81.1%

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

      1. clear-num 81.1%

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

      2. un-div-inv 82.1%

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

      3. clear-num 82.1%

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

      4. un-div-inv 82.0%

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

      5. associate-/r/ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. associate-*l/ 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l/ 82.0%

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

      4. associate-/l/ 82.1%

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

      5. associate-/r/ 82.1%

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

   8. Simplified 82.1%

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

      1. unpow2 82.1%

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

      2. div-inv 82.1%

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

      3. times-frac 92.6%

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

      4. associate-*l/ 92.6%

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

      5. associate-*l/ 92.6%

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

   10. Applied egg-rr 92.6%

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

       1. associate-*l/ 92.6%

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

       2. associate-/l* 91.7%

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

       3. associate-*l/ 91.7%

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

       4. *-commutative 91.7%

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

       5. associate-/r/ 91.7%

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

       6. /-rgt-identity 91.7%

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

       7. associate-*l/ 91.7%

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

       8. associate-/l* 91.7%

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

       9. associate-*l/ 91.7%

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

       10. *-commutative 91.7%

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

   12. Simplified 91.7%

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

       1. associate-*l/ 91.7%

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

       2. div-inv 91.7%

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

       3. metadata-eval 91.7%

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

   14. Applied egg-rr 91.7%

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

       1. associate-/l* 91.7%

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

       2. *-commutative 91.7%

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

   16. Simplified 91.7%

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

       1. div-inv 91.7%

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

       2. frac-times 92.6%

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

   18. Applied egg-rr 92.6%

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

       1. associate-*r/ 92.6%

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

       2. associate-/l* 92.6%

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

       3. *-commutative 92.6%

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

       4. associate-/l/ 92.6%

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

       5. *-commutative 92.6%

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

       6. *-rgt-identity 92.6%

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

       7. associate-*l/ 92.6%

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

       8. associate-*r/ 92.6%

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

       9. associate-/l* 88.8%

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

   20. Simplified 88.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-183}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D}{2}}{\ell} \cdot \left(0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{\frac{M}{2}}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.8× speedup

Math

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

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
    (* (* (/ D_m d) (/ (/ M_m 2.0) l)) (* h (* M_m (/ (* D_m 0.5) d))))))))

C

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

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 - (((d_m / d) * ((m_m / 2.0d0) / l)) * (h * (m_m * ((d_m * 0.5d0) / d))))))
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 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))));
}

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 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))))

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(Float64(D_m / d) * Float64(Float64(M_m / 2.0) / l)) * Float64(h * Float64(M_m * Float64(Float64(D_m * 0.5) / d)))))))
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 - (((D_m / d) * ((M_m / 2.0) / l)) * (h * (M_m * ((D_m * 0.5) / d))))));
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[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m / 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

3. Simplified 79.0%

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

   1. clear-num 79.0%

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

   2. un-div-inv 80.0%

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

   3. clear-num 80.0%

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

   4. un-div-inv 80.4%

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

   5. associate-/r/ 80.4%

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

6. Applied egg-rr 80.4%

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

   1. associate-*l/ 80.4%

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

   2. *-commutative 80.4%

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

   3. associate-*l/ 80.4%

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

   4. associate-/l/ 80.5%

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

   5. associate-/r/ 80.5%

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

8. Simplified 80.5%

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

   1. unpow2 80.5%

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

   2. div-inv 80.5%

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

   3. times-frac 90.3%

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

   4. associate-*l/ 90.3%

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

   5. associate-*l/ 90.3%

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

10. Applied egg-rr 90.3%

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

    1. associate-*l/ 89.5%

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

    2. associate-/l* 88.1%

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

    3. associate-*l/ 88.1%

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

    4. *-commutative 88.1%

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

    5. associate-/r/ 88.1%

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

    6. /-rgt-identity 88.1%

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

    7. associate-*l/ 88.8%

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

    8. associate-/l* 89.6%

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

    9. associate-*l/ 89.6%

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

    10. *-commutative 89.6%

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

12. Simplified 89.6%

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

    1. associate-*l/ 88.8%

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

    2. div-inv 88.8%

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

    3. metadata-eval 88.8%

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

14. Applied egg-rr 88.8%

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

    1. associate-/l* 88.1%

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

    2. *-commutative 88.1%

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

16. Simplified 88.1%

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

    1. div-inv 88.1%

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

    2. frac-times 89.6%

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

18. Applied egg-rr 89.6%

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

    1. associate-*r/ 89.6%

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

    2. associate-/l* 90.3%

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

    3. *-commutative 90.3%

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

    4. associate-/l/ 90.3%

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

    5. *-commutative 90.3%

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

    6. *-rgt-identity 90.3%

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

    7. associate-*l/ 90.3%

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

    8. associate-*r/ 90.3%

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

    9. associate-/l* 86.4%

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

20. Simplified 86.4%

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

21. Final simplification 86.4%

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{\frac{M}{2}}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)} \]
22. Add Preprocessing

Alternative 4: 68.7% 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 79.4%

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

3. Simplified 79.0%

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

5. Taylor expanded in D around 0 68.8%

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

6. Final simplification 68.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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))))))