Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 88.9%

Time: 14.3s

Alternatives: 3

Speedup: 1.8×

• Rival profile vector overflowed, profile may not be complete

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: 80.4% 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: 88.9% accurate, 1.8× speedup

Math

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

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 (sqrt (- 1.0 (* h (/ t_0 (/ l t_0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((0.5 * M_m) / d);
	return w0 * sqrt((1.0 - (h * (t_0 / (l / t_0)))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d_m * ((0.5d0 * m_m) / d)
    code = w0 * sqrt((1.0d0 - (h * (t_0 / (l / t_0)))))
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.sqrt((1.0 - (h * (t_0 / (l / t_0)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * ((0.5 * M_m) / d)
	return w0 * math.sqrt((1.0 - (h * (t_0 / (l / t_0)))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(Float64(0.5 * M_m) / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 / Float64(l / t_0))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * ((0.5 * M_m) / d);
	tmp = w0 * sqrt((1.0 - (h * (t_0 / (l / t_0)))));
end

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Simplified 78.3%

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

6. Applied egg-rr 78.3%

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

   1. unpow1 78.3%

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

   2. associate-*l/ 85.0%

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

   3. associate-/l* 85.3%

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

   4. associate-*r/ 85.0%

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

   5. times-frac 85.3%

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

   6. associate-*r/ 85.0%

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

8. Simplified 85.0%

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

   1. unpow2 85.0%

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

   2. *-un-lft-identity 85.0%

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

   3. times-frac 86.5%

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

   4. associate-/l* 85.3%

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

   5. frac-times 86.5%

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

   6. associate-*r/ 85.3%

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

   7. *-un-lft-identity 85.3%

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

   8. times-frac 85.3%

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

   9. metadata-eval 85.3%

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

   10. associate-/l* 86.8%

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

   11. frac-times 85.3%

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

   12. associate-*r/ 86.9%

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

   13. *-un-lft-identity 86.9%

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

   14. times-frac 86.8%

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

   15. metadata-eval 86.8%

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

10. Applied egg-rr 86.8%

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

    1. clear-num 86.8%

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

    2. frac-times 86.8%

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

    3. metadata-eval 86.8%

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

    4. div-inv 86.8%

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

    5. /-rgt-identity 86.8%

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

    6. associate-*r/ 86.9%

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

    7. metadata-eval 86.9%

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

    8. times-frac 86.9%

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

    9. *-un-lft-identity 86.9%

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

    10. *-un-lft-identity 86.9%

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

    11. associate-*r/ 86.8%

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

12. Applied egg-rr 86.8%

    \[\leadsto w0 \cdot \sqrt{1 - h \cdot \color{blue}{\frac{D \cdot \frac{0.5 \cdot M}{d}}{\frac{\ell}{D \cdot \frac{0.5 \cdot M}{d}}}}} \]
13. Add Preprocessing

Alternative 2: 86.3% accurate, 1.8× speedup

Math

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

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(h * Float64(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[(h * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.0%

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

3. Simplified 78.3%

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

6. Applied egg-rr 78.3%

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

   1. unpow1 78.3%

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

   2. associate-*l/ 85.0%

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

   3. associate-/l* 85.3%

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

   4. associate-*r/ 85.0%

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

   5. times-frac 85.3%

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

   6. associate-*r/ 85.0%

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

8. Simplified 85.0%

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

   1. associate-/l* 85.3%

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

   2. frac-times 85.0%

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

   3. associate-*r/ 85.3%

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

   4. unpow-prod-down 70.4%

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

   5. add-sqr-sqrt 70.4%

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

   6. unpow-prod-down 70.4%

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

   7. sqrt-pow1 62.5%

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

   8. metadata-eval 62.5%

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

   9. pow1 62.5%

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

   10. *-un-lft-identity 62.5%

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

   11. times-frac 62.5%

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

   12. metadata-eval 62.5%

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

   13. unpow-prod-down 75.2%

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

   14. sqrt-pow1 85.3%

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

   15. metadata-eval 85.3%

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

   16. pow1 85.3%

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

   17. *-un-lft-identity 85.3%

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

   18. times-frac 85.3%

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

   19. metadata-eval 85.3%

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

10. Applied egg-rr 85.3%

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

    1. pow2 85.3%

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

    2. associate-*r* 85.3%

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

    3. metadata-eval 85.3%

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

    4. div-inv 85.3%

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

    5. unpow2 85.3%

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

    6. associate-*l/ 85.3%

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

    7. associate-*l/ 85.3%

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

    8. frac-times 85.0%

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

    9. metadata-eval 85.0%

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

12. Applied egg-rr 85.0%

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

Alternative 3: 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 78.0%

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

3. Simplified 78.3%

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

5. Taylor expanded in D around 0 66.8%

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

Reproduce

herbie shell --seed 2024155 
(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))))))