Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.9%

Time: 16.9s

Alternatives: 5

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: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\\ w0 \cdot {\left(1 - \left(h \cdot t\_0\right) \cdot \frac{t\_0}{\ell}\right)}^{0.5} \end{array} \end{array} \]

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)))))
   (* w0 (pow (- 1.0 (* (* h t_0) (/ t_0 l))) 0.5))))

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));
	return w0 * pow((1.0 - ((h * t_0) * (t_0 / l))), 0.5);
}

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 * (0.5d0 * (m_m / d))
    code = w0 * ((1.0d0 - ((h * t_0) * (t_0 / l))) ** 0.5d0)
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));
	return w0 * Math.pow((1.0 - ((h * t_0) * (t_0 / l))), 0.5);
}

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))
	return w0 * math.pow((1.0 - ((h * t_0) * (t_0 / l))), 0.5)

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)))
	return Float64(w0 * (Float64(1.0 - Float64(Float64(h * t_0) * Float64(t_0 / l))) ^ 0.5))
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 * (0.5 * (M_m / d));
	tmp = w0 * ((1.0 - ((h * t_0) * (t_0 / l))) ^ 0.5);
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]}, N[(w0 * N[Power[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.9%

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

3. Simplified 81.0%

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

6. Applied egg-rr 79.9%

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

   1. *-lft-identity 79.9%

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

   2. associate-*l/ 84.7%

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

   3. associate-/l* 85.4%

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

   4. associate-/l* 85.8%

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

8. Simplified 85.8%

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

   1. unpow2 85.8%

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

   2. *-un-lft-identity 85.8%

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

   3. times-frac 88.0%

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

   4. *-un-lft-identity 88.0%

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

   5. times-frac 88.0%

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

   6. metadata-eval 88.0%

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

   7. *-un-lft-identity 88.0%

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

   8. times-frac 88.0%

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

   9. metadata-eval 88.0%

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

10. Applied egg-rr 88.0%

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

    1. associate-/l* 85.0%

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

    2. associate-*r/ 85.0%

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

12. Applied egg-rr 85.0%

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

    1. pow1/2 85.0%

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

    2. /-rgt-identity 85.0%

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

    3. associate-*r* 86.5%

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

    4. associate-*r/ 89.2%

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

    5. associate-*r/ 89.2%

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

14. Applied egg-rr 89.2%

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

Alternative 2: 71.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.25 \cdot 10^{+43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{w0 \cdot \frac{h}{\ell}}{{d}^{2}}\right)\\ \end{array} \end{array} \]

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 2.25e+43)
   w0
   (+
    w0
    (*
     -0.125
     (* (* (* D_m M_m) (* D_m M_m)) (/ (* w0 (/ h l)) (pow d 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e+43) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * (h / l)) / pow(d, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2.25d+43) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d_m * m_m) * (d_m * m_m)) * ((w0 * (h / l)) / (d ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e+43) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * (h / l)) / Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2.25e+43:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * (h / l)) / math.pow(d, 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.25e+43)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) * Float64(Float64(w0 * Float64(h / l)) / (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.25e+43)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * (h / l)) / (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.25e+43], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h / l), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.25e43

   1. Initial program 82.4%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in D around 0 78.0%

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

   if 2.25e43 < M

   1. Initial program 70.3%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in D around 0 32.2%

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

      1. associate-/l* 33.8%

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

      2. associate-/l* 39.6%

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

   7. Simplified 39.6%

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

      1. pow1 39.6%

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

      2. associate-*r* 39.5%

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

      3. pow-prod-down 47.6%

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

      4. *-commutative 47.6%

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

      5. times-frac 55.4%

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

   9. Applied egg-rr 55.4%

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

       1. unpow1 55.4%

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

       2. associate-*r/ 53.5%

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

   11. Simplified 53.5%

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

       1. unpow2 53.5%

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

   13. Applied egg-rr 53.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{+43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{w0 \cdot \frac{h}{\ell}}{{d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7200000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \end{array} \end{array} \]

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 7200000.0)
   w0
   (+
    w0
    (*
     -0.125
     (* (* (* D_m M_m) (* D_m M_m)) (/ (* w0 h) (* l (pow d 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7200000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * h) / (l * pow(d, 2.0)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 7200000.0d0) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d_m * m_m) * (d_m * m_m)) * ((w0 * h) / (l * (d ** 2.0d0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7200000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * h) / (l * Math.pow(d, 2.0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 7200000.0:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * h) / (l * math.pow(d, 2.0)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 7200000.0)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) * Float64(Float64(w0 * h) / Float64(l * (d ^ 2.0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 7200000.0)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D_m * M_m) * (D_m * M_m)) * ((w0 * h) / (l * (d ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 7200000.0], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2e6

   1. Initial program 82.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in D around 0 77.4%

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

   if 7.2e6 < M

   1. Initial program 71.9%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in D around 0 38.3%

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

      1. associate-/l* 39.7%

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

      2. associate-/l* 44.6%

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

   7. Simplified 44.6%

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

      1. pow1 44.6%

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

      2. associate-*r* 44.5%

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

      3. pow-prod-down 51.2%

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

      4. *-commutative 51.2%

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

      5. times-frac 57.8%

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

   9. Applied egg-rr 57.8%

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

       1. unpow1 57.8%

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

       2. associate-*r/ 56.2%

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

   11. Simplified 56.2%

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

       1. unpow2 56.2%

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

   13. Applied egg-rr 56.2%

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

   14. Taylor expanded in h around 0 51.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7200000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{M\_m}{d}\\ w0 \cdot \sqrt{1 - \left(h \cdot \left(D\_m \cdot t\_0\right)\right) \cdot \left(D\_m \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

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 (* 0.5 (/ M_m d))))
   (* w0 (sqrt (- 1.0 (* (* h (* D_m t_0)) (* D_m (/ 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 = 0.5 * (M_m / d);
	return w0 * sqrt((1.0 - ((h * (D_m * t_0)) * (D_m * (t_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 = 0.5d0 * (m_m / d)
    code = w0 * sqrt((1.0d0 - ((h * (d_m * t_0)) * (d_m * (t_0 / 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 = 0.5 * (M_m / d);
	return w0 * Math.sqrt((1.0 - ((h * (D_m * t_0)) * (D_m * (t_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 = 0.5 * (M_m / d)
	return w0 * math.sqrt((1.0 - ((h * (D_m * t_0)) * (D_m * (t_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(0.5 * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(D_m * t_0)) * Float64(D_m * Float64(t_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 = 0.5 * (M_m / d);
	tmp = w0 * sqrt((1.0 - ((h * (D_m * t_0)) * (D_m * (t_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[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(D$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.9%

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

3. Simplified 81.0%

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

6. Applied egg-rr 79.9%

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

   1. *-lft-identity 79.9%

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

   2. associate-*l/ 84.7%

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

   3. associate-/l* 85.4%

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

   4. associate-/l* 85.8%

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

8. Simplified 85.8%

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

   1. unpow2 85.8%

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

   2. *-un-lft-identity 85.8%

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

   3. times-frac 88.0%

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

   4. *-un-lft-identity 88.0%

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

   5. times-frac 88.0%

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

   6. metadata-eval 88.0%

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

   7. *-un-lft-identity 88.0%

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

   8. times-frac 88.0%

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

   9. metadata-eval 88.0%

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

10. Applied egg-rr 88.0%

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

    1. associate-/l* 85.0%

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

    2. associate-*r/ 85.0%

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

12. Applied egg-rr 85.0%

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

    1. *-un-lft-identity 85.0%

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

    2. /-rgt-identity 85.0%

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

    3. associate-*r* 86.4%

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

    4. associate-*r/ 89.1%

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

    5. associate-*r/ 89.1%

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

14. Applied egg-rr 89.1%

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

    1. *-lft-identity 89.1%

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

    2. *-commutative 89.1%

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

    3. associate-*r/ 86.4%

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

16. Simplified 86.4%

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

17. Final simplification 86.4%

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

Alternative 5: 67.6% 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 79.9%

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

3. Simplified 81.0%

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

5. Taylor expanded in D around 0 70.6%

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

Reproduce

herbie shell --seed 2024092 
(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))))))