Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.6%

Time: 16.4s

Alternatives: 6

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: 81.4% 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: 89.6% accurate, 1.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* M (/ 0.5 d)))))
   (* w0 (sqrt (- 1.0 (* t_0 (/ (* h t_0) l)))))))

C

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

Fortran

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

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	return w0 * Math.sqrt((1.0 - (t_0 * ((h * t_0) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l/ 85.0%

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

   5. associate-*l/ 84.7%

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

   6. *-commutative 84.7%

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

   7. div-inv 84.6%

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

   8. associate-/r* 84.6%

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

   9. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. associate-*l/ 79.1%

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

   2. unpow2 79.1%

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

   3. associate-*r* 80.7%

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

8. Applied egg-rr 80.7%

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

   1. associate-*l/ 87.3%

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

10. Applied egg-rr 87.3%

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

11. Final simplification 87.3%

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

Alternative 2: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;d \leq 2 \cdot 10^{-143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D\_m}{d} \cdot \left(0.5 \cdot \frac{h \cdot M}{\ell}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* M (/ 0.5 d)))))
   (if (<= d 2e-143)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     (*
      w0
      (sqrt
       (-
        1.0
        (* (* (/ D_m d) (* 0.5 (/ (* h M) l))) (* (/ M 2.0) (/ D_m d)))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	double tmp;
	if (d <= 2e-143) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m * (0.5d0 / d))
    if (d <= 2d-143) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m / d) * (0.5d0 * ((h * m) / l))) * ((m / 2.0d0) * (d_m / d)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	double tmp;
	if (d <= 2e-143) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = D_m * (M * (0.5 / d))
	tmp = 0
	if d <= 2e-143:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M * Float64(0.5 / d)))
	tmp = 0.0
	if (d <= 2e-143)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(0.5 * Float64(Float64(h * M) / l))) * Float64(Float64(M / 2.0) * Float64(D_m / d))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = D_m * (M * (0.5 / d));
	tmp = 0.0;
	if (d <= 2e-143)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0 * sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2e-143], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * N[(N[(h * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9999999999999999e-143

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. frac-times 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-*l/ 82.4%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. div-inv 82.3%

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

      8. associate-/r* 82.3%

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

      9. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*l/ 76.5%

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

      2. unpow2 76.5%

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

      3. associate-*r* 77.2%

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

   8. Applied egg-rr 77.2%

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

   if 1.9999999999999999e-143 < d

   1. Initial program 84.1%

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

   3. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. frac-times 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*l/ 89.2%

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

      5. associate-*l/ 88.3%

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

      6. *-commutative 88.3%

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

      7. div-inv 88.3%

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

      8. associate-/r* 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-*l/ 83.2%

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

      2. unpow2 83.2%

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

      3. associate-*r* 86.2%

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

   8. Applied egg-rr 86.2%

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

   9. Taylor expanded in h around 0 87.8%

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

       1. *-commutative 87.8%

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

       2. times-frac 86.2%

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

       3. associate-*l* 86.2%

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

   11. Simplified 86.2%

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

       1. associate-*r* 86.1%

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

       2. clear-num 86.1%

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

       3. un-div-inv 86.1%

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

       4. div-inv 86.1%

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

       5. metadata-eval 86.1%

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

   13. Applied egg-rr 86.1%

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

       1. times-frac 88.1%

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

   15. Simplified 88.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h \cdot M}{\ell}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 10^{-143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{D\_m}{\frac{d}{\frac{M}{2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D\_m}{d} \cdot \left(0.5 \cdot \frac{h \cdot M}{\ell}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= d 1e-143)
   (*
    w0
    (sqrt
     (- 1.0 (* (* D_m (* M (/ 0.5 d))) (* (/ h l) (/ D_m (/ d (/ M 2.0))))))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ D_m d) (* 0.5 (/ (* h M) l))) (* (/ M 2.0) (/ D_m d))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1e-143) {
		tmp = w0 * sqrt((1.0 - ((D_m * (M * (0.5 / d))) * ((h / l) * (D_m / (d / (M / 2.0)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 1d-143) then
        tmp = w0 * sqrt((1.0d0 - ((d_m * (m * (0.5d0 / d))) * ((h / l) * (d_m / (d / (m / 2.0d0)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m / d) * (0.5d0 * ((h * m) / l))) * ((m / 2.0d0) * (d_m / d)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1e-143) {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * (M * (0.5 / d))) * ((h / l) * (D_m / (d / (M / 2.0)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if d <= 1e-143:
		tmp = w0 * math.sqrt((1.0 - ((D_m * (M * (0.5 / d))) * ((h / l) * (D_m / (d / (M / 2.0)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (d <= 1e-143)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(M * Float64(0.5 / d))) * Float64(Float64(h / l) * Float64(D_m / Float64(d / Float64(M / 2.0))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(0.5 * Float64(Float64(h * M) / l))) * Float64(Float64(M / 2.0) * Float64(D_m / d))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 1e-143)
		tmp = w0 * sqrt((1.0 - ((D_m * (M * (0.5 / d))) * ((h / l) * (D_m / (d / (M / 2.0)))))));
	else
		tmp = w0 * sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 1e-143], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(D$95$m / N[(d / N[(M / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * N[(N[(h * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.9999999999999995e-144

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. frac-times 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-*l/ 82.4%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. div-inv 82.3%

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

      8. associate-/r* 82.3%

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

      9. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*l/ 76.5%

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

      2. unpow2 76.5%

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

      3. associate-*r* 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. associate-*r* 74.2%

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

      2. clear-num 74.2%

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

      3. un-div-inv 74.2%

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

      4. div-inv 74.2%

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

      5. metadata-eval 74.2%

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

   10. Applied egg-rr 76.0%

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

       1. associate-/l* 77.3%

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

       2. associate-/l* 77.3%

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

   12. Simplified 77.3%

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

   if 9.9999999999999995e-144 < d

   1. Initial program 84.1%

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

   3. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. frac-times 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*l/ 89.2%

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

      5. associate-*l/ 88.3%

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

      6. *-commutative 88.3%

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

      7. div-inv 88.3%

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

      8. associate-/r* 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-*l/ 83.2%

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

      2. unpow2 83.2%

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

      3. associate-*r* 86.2%

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

   8. Applied egg-rr 86.2%

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

   9. Taylor expanded in h around 0 87.8%

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

       1. *-commutative 87.8%

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

       2. times-frac 86.2%

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

       3. associate-*l* 86.2%

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

   11. Simplified 86.2%

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

       1. associate-*r* 86.1%

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

       2. clear-num 86.1%

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

       3. un-div-inv 86.1%

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

       4. div-inv 86.1%

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

       5. metadata-eval 86.1%

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

   13. Applied egg-rr 86.1%

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

       1. times-frac 88.1%

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

   15. Simplified 88.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{-143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{D}{\frac{d}{\frac{M}{2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \left(0.5 \cdot \frac{h \cdot M}{\ell}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 1.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (* (* D_m (* M (/ 0.5 d))) (* (/ D_m d) (* 0.5 (/ (* h M) l))))))))

C

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

Fortran

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

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((D_m * (M * (0.5 / d))) * ((D_m / d) * (0.5 * ((h * M) / l))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l/ 85.0%

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

   5. associate-*l/ 84.7%

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

   6. *-commutative 84.7%

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

   7. div-inv 84.6%

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

   8. associate-/r* 84.6%

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

   9. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. associate-*l/ 79.1%

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

   2. unpow2 79.1%

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

   3. associate-*r* 80.7%

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

8. Applied egg-rr 80.7%

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

9. Taylor expanded in h around 0 79.9%

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

    1. *-commutative 79.9%

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

    2. times-frac 79.2%

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

    3. associate-*l* 79.2%

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

11. Simplified 79.2%

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

12. Final simplification 79.2%

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

Alternative 5: 84.4% accurate, 1.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (* (* (/ D_m d) (* 0.5 (/ (* h M) l))) (* (/ M 2.0) (/ D_m d)))))))

C

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

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * sqrt((1.0d0 - (((d_m / d) * (0.5d0 * ((h * m) / l))) * ((m / 2.0d0) * (d_m / d)))))
end function

Java

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

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))))

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(0.5 * Float64(Float64(h * M) / l))) * Float64(Float64(M / 2.0) * Float64(D_m / d))))))
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - (((D_m / d) * (0.5 * ((h * M) / l))) * ((M / 2.0) * (D_m / d)))));
end

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l/ 85.0%

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

   5. associate-*l/ 84.7%

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

   6. *-commutative 84.7%

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

   7. div-inv 84.6%

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

   8. associate-/r* 84.6%

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

   9. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. associate-*l/ 79.1%

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

   2. unpow2 79.1%

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

   3. associate-*r* 80.7%

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

8. Applied egg-rr 80.7%

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

9. Taylor expanded in h around 0 79.9%

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

    1. *-commutative 79.9%

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

    2. times-frac 79.2%

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

    3. associate-*l* 79.2%

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

11. Simplified 79.2%

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

    1. associate-*r* 78.8%

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

    2. clear-num 78.8%

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

    3. un-div-inv 78.8%

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

    4. div-inv 78.8%

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

    5. metadata-eval 78.8%

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

13. Applied egg-rr 78.8%

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

    1. times-frac 80.3%

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

15. Simplified 80.3%

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

16. Final simplification 80.3%

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

Alternative 6: 68.7% accurate, 216.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d) :precision binary64 w0)

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return w0
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

5. Taylor expanded in D around 0 68.2%

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

6. Final simplification 68.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))