Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.2%

Time: 16.5s

Alternatives: 7

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 1.8× speedup

Math

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

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 81.0%

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

3. Simplified 81.7%

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

   1. *-commutative 81.7%

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

   2. unpow2 81.7%

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

   3. associate-*r* 83.3%

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

   4. *-commutative 83.3%

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

   5. associate-/r* 83.3%

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

   6. associate-*l/ 81.4%

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

   7. associate-*r/ 83.3%

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

   8. *-commutative 83.3%

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

   9. associate-/r* 83.3%

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

   10. associate-*l/ 82.5%

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

   11. associate-*r/ 84.5%

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

   12. associate-*r* 82.5%

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

   13. unpow2 82.5%

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

   14. associate-*l/ 88.1%

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

6. Applied egg-rr 87.3%

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

   1. associate-*r/ 87.0%

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

   2. frac-times 87.3%

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

   3. unpow2 87.3%

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

   4. associate-*l/ 87.3%

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

   5. associate-*l/ 87.3%

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

   6. frac-times 87.3%

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

   7. metadata-eval 87.3%

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

8. Applied egg-rr 87.3%

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

Alternative 2: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-293}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-293)
   (+ w0 (* -0.125 (* (pow (* D_m (/ M_m d)) 2.0) (* h (/ w0 l)))))
   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) {
	double tmp;
	if ((h / l) <= -5e-293) {
		tmp = w0 + (-0.125 * (pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-5d-293)) then
        tmp = w0 + ((-0.125d0) * (((d_m * (m_m / d)) ** 2.0d0) * (h * (w0 / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-293) {
		tmp = w0 + (-0.125 * (Math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e-293:
		tmp = w0 + (-0.125 * (math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-293)
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(h * Float64(w0 / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-293)
		tmp = w0 + (-0.125 * (((D_m * (M_m / d)) ^ 2.0) * (h * (w0 / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-293], N[(w0 + N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000003e-293

   1. Initial program 79.1%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in D around 0 42.8%

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

      1. associate-/l* 43.5%

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

      2. times-frac 44.1%

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

      3. *-commutative 44.1%

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

   7. Simplified 44.1%

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

      1. associate-*r/ 47.0%

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

      2. unpow2 47.0%

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

      3. unpow2 47.0%

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

      4. frac-times 55.4%

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

      5. pow2 55.4%

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

      6. *-commutative 55.4%

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

   9. Applied egg-rr 55.4%

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

   10. Taylor expanded in D around 0 42.8%

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

       1. associate-*r* 44.2%

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

       2. times-frac 44.8%

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

       3. associate-/l* 44.1%

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

       4. unpow2 44.1%

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

       5. unpow2 44.1%

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

       6. unpow2 44.1%

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

       7. times-frac 53.3%

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

       8. swap-sqr 62.1%

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

       9. unpow2 62.1%

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

       10. associate-*r/ 64.9%

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

   12. Simplified 64.9%

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

   if -5.0000000000000003e-293 < (/.f64 h l)

   1. Initial program 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in D around 0 94.6%

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

Alternative 3: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-293}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(M\_m \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-293)
   (+ w0 (* -0.125 (* (* h (/ w0 l)) (pow (* M_m (/ D_m d)) 2.0))))
   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) {
	double tmp;
	if ((h / l) <= -5e-293) {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * pow((M_m * (D_m / d)), 2.0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-5d-293)) then
        tmp = w0 + ((-0.125d0) * ((h * (w0 / l)) * ((m_m * (d_m / d)) ** 2.0d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-293) {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * Math.pow((M_m * (D_m / d)), 2.0)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e-293:
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * math.pow((M_m * (D_m / d)), 2.0)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-293)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 / l)) * (Float64(M_m * Float64(D_m / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-293)
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * ((M_m * (D_m / d)) ^ 2.0)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-293], N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000003e-293

   1. Initial program 79.1%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in D around 0 42.8%

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

      1. associate-/l* 43.5%

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

      2. times-frac 44.1%

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

      3. *-commutative 44.1%

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

   7. Simplified 44.1%

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

      1. unpow2 44.1%

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

      2. unpow2 44.1%

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

      3. frac-times 53.3%

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

   9. Applied egg-rr 53.3%

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

       1. unpow2 53.3%

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

   11. Applied egg-rr 53.3%

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

   12. Taylor expanded in D around 0 42.8%

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

       1. associate-*r* 44.2%

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

       2. *-commutative 44.2%

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

       3. times-frac 44.8%

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

       4. associate-/l* 44.1%

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

       5. unpow2 44.1%

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

       6. unpow2 44.1%

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

       7. unpow2 44.1%

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

       8. times-frac 53.3%

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

       9. swap-sqr 62.1%

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

       10. associate-/l* 62.1%

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

       11. associate-/l* 62.1%

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

       12. unpow2 62.1%

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

       13. associate-*r/ 64.9%

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

       14. *-commutative 64.9%

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

       15. associate-*r/ 62.1%

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

       16. *-commutative 62.1%

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

       17. associate-/l* 64.9%

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

       18. *-commutative 64.9%

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

       19. associate-/l* 64.9%

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

   14. Simplified 64.9%

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

   if -5.0000000000000003e-293 < (/.f64 h l)

   1. Initial program 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in D around 0 94.6%

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

Alternative 4: 84.8% accurate, 1.8× speedup

Math

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

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 (/ (* M_m (* D_m (/ M_m d))) (/ (* 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 * ((M_m * (D_m * (M_m / d))) / ((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 * ((m_m * (d_m * (m_m / d))) / ((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 * ((M_m * (D_m * (M_m / d))) / ((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 * ((M_m * (D_m * (M_m / d))) / ((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(Float64(h * Float64(Float64(M_m * Float64(D_m * Float64(M_m / d))) / 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 * ((M_m * (D_m * (M_m / d))) / ((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[(N[(h * N[(N[(M$95$m * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

   1. *-commutative 81.7%

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

   2. unpow2 81.7%

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

   3. associate-*r* 83.3%

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

   4. *-commutative 83.3%

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

   5. associate-/r* 83.3%

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

   6. associate-*l/ 81.4%

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

   7. associate-*r/ 83.3%

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

   8. *-commutative 83.3%

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

   9. associate-/r* 83.3%

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

   10. associate-*l/ 82.5%

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

   11. associate-*r/ 84.5%

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

   12. associate-*r* 82.5%

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

   13. unpow2 82.5%

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

   14. associate-*l/ 88.1%

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

6. Applied egg-rr 87.3%

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

   1. unpow2 87.3%

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

   2. associate-*r/ 86.2%

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

   3. *-commutative 86.2%

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

   4. associate-/r/ 87.0%

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

   5. frac-times 87.3%

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

   6. associate-*l/ 87.3%

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

   7. frac-times 86.6%

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

   8. associate-/l* 86.6%

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

8. Applied egg-rr 86.6%

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

9. Taylor expanded in d around 0 86.6%

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

    1. associate-*r/ 86.6%

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

11. Simplified 86.6%

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

12. Final simplification 86.6%

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

Alternative 5: 85.3% accurate, 1.8× speedup

Math

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

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(h * Float64(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[(h * N[(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] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

   1. *-commutative 81.7%

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

   2. unpow2 81.7%

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

   3. associate-*r* 83.3%

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

   4. *-commutative 83.3%

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

   5. associate-/r* 83.3%

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

   6. associate-*l/ 81.4%

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

   7. associate-*r/ 83.3%

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

   8. *-commutative 83.3%

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

   9. associate-/r* 83.3%

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

   10. associate-*l/ 82.5%

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

   11. associate-*r/ 84.5%

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

   12. associate-*r* 82.5%

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

   13. unpow2 82.5%

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

   14. associate-*l/ 88.1%

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

6. Applied egg-rr 87.3%

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

   1. unpow2 87.3%

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

   2. associate-*r/ 86.2%

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

   3. *-commutative 86.2%

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

   4. associate-/r/ 87.0%

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

   5. frac-times 87.3%

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

   6. associate-*l/ 87.3%

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

   7. frac-times 86.6%

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

   8. associate-/l* 86.6%

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

8. Applied egg-rr 86.6%

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

   1. associate-/l* 86.2%

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

   2. associate-/l* 86.9%

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

   3. associate-*r* 86.9%

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

   4. metadata-eval 86.9%

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

10. Applied egg-rr 86.9%

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

Alternative 6: 65.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 1.2e-22)
   (+ w0 (* -0.125 (* (* D_m D_m) (* (* h (/ w0 l)) (* (/ M_m d) (/ M_m d))))))
   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) {
	double tmp;
	if (d <= 1.2e-22) {
		tmp = w0 + (-0.125 * ((D_m * D_m) * ((h * (w0 / l)) * ((M_m / d) * (M_m / d)))));
	} else {
		tmp = w0;
	}
	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 <= 1.2d-22) then
        tmp = w0 + ((-0.125d0) * ((d_m * d_m) * ((h * (w0 / l)) * ((m_m / d) * (m_m / d)))))
    else
        tmp = w0
    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 <= 1.2e-22) {
		tmp = w0 + (-0.125 * ((D_m * D_m) * ((h * (w0 / l)) * ((M_m / d) * (M_m / d)))));
	} else {
		tmp = w0;
	}
	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 <= 1.2e-22:
		tmp = w0 + (-0.125 * ((D_m * D_m) * ((h * (w0 / l)) * ((M_m / d) * (M_m / d)))))
	else:
		tmp = w0
	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 <= 1.2e-22)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D_m * D_m) * Float64(Float64(h * Float64(w0 / l)) * Float64(Float64(M_m / d) * Float64(M_m / d))))));
	else
		tmp = w0;
	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 <= 1.2e-22)
		tmp = w0 + (-0.125 * ((D_m * D_m) * ((h * (w0 / l)) * ((M_m / d) * (M_m / d)))));
	else
		tmp = w0;
	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, 1.2e-22], N[(w0 + N[(-0.125 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.20000000000000001e-22

   1. Initial program 77.9%

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

   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. Taylor expanded in D around 0 46.4%

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

      1. associate-/l* 46.3%

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

      2. times-frac 45.7%

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

      3. *-commutative 45.7%

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

   7. Simplified 45.7%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. frac-times 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. unpow2 58.6%

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

   11. Applied egg-rr 58.6%

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

   12. Taylor expanded in w0 around 0 58.6%

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

       1. associate-/l* 59.7%

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

   14. Simplified 59.7%

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

   if 1.20000000000000001e-22 < d

   1. Initial program 88.5%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in D around 0 80.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% 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 81.0%

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

3. Simplified 81.7%

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

5. Taylor expanded in D around 0 66.4%

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

Reproduce

herbie shell --seed 2024152 
(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))))))