Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.3% → 88.5%

Time: 15.1s

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: 80.3% 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: 88.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 - (((((0.5d0 * m) / d_1) * d) * h) / (((d_1 * (2.0d0 / m)) / d) * l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 82.3%

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

   1. *-commutative 82.3%

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

   2. associate-*l/ 82.5%

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

   3. associate-*r/ 81.7%

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

   4. times-frac 82.5%

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

   5. unpow2 82.5%

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

   6. clear-num 82.5%

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

   7. clear-num 82.5%

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

   8. frac-times 82.5%

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

   9. metadata-eval 82.5%

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

   10. times-frac 81.3%

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

   11. times-frac 82.1%

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

6. Applied egg-rr 82.1%

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

   1. associate-/r* 82.1%

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

   2. associate-/r* 82.1%

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

8. Simplified 82.1%

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

   1. frac-times 90.1%

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

   2. associate-/r/ 89.6%

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

   3. associate-/r/ 89.6%

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

   4. metadata-eval 89.6%

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

10. Applied egg-rr 89.6%

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

    1. associate-*r/ 90.3%

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

12. Applied egg-rr 90.3%

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

13. Final simplification 90.3%

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

Alternative 2: 78.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{\frac{d}{M \cdot D}} \cdot \left(D \cdot \left(M \cdot \frac{h}{d}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 3.4e-40)
   w0
   (* w0 (sqrt (- 1.0 (/ (* (/ 0.25 (/ d (* M D))) (* D (* M (/ h d)))) l))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.4e-40) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (((0.25 / (d / (M * D))) * (D * (M * (h / d)))) / l)));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= 3.4d-40) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (((0.25d0 / (d_1 / (m * d))) * (d * (m * (h / d_1)))) / l)))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.4e-40) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((0.25 / (d / (M * D))) * (D * (M * (h / d)))) / l)));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 3.4e-40:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (((0.25 / (d / (M * D))) * (D * (M * (h / d)))) / l)))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 3.4e-40)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.25 / Float64(d / Float64(M * D))) * Float64(D * Float64(M * Float64(h / d)))) / l))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 3.4e-40)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (((0.25 / (d / (M * D))) * (D * (M * (h / d)))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 3.4e-40], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.25 / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.39999999999999984e-40

   1. Initial program 85.1%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in D around 0 74.4%

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

   if 3.39999999999999984e-40 < D

   1. Initial program 74.0%

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

   3. Simplified 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 74.0%

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

      3. associate-*r/ 69.3%

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

      4. times-frac 74.0%

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

      5. unpow2 74.0%

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

      6. associate-/r* 74.0%

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

      7. frac-times 62.3%

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

      8. div-inv 62.3%

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

      9. *-commutative 62.3%

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

      10. metadata-eval 62.3%

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

      11. associate-*r* 62.3%

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

      12. *-commutative 62.3%

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

   6. Applied egg-rr 62.3%

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

      1. associate-*r/ 64.4%

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

      2. times-frac 74.7%

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

      3. *-commutative 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-*l/ 74.7%

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

      6. associate-*r/ 74.7%

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

      7. *-commutative 74.7%

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

      8. *-commutative 74.7%

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

      9. *-commutative 74.7%

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

   8. Applied egg-rr 74.7%

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

      1. pow1 74.7%

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

      2. associate-*l* 79.2%

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

      3. associate-*r* 79.2%

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

      4. associate-/l* 76.1%

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

   10. Applied egg-rr 76.1%

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

       1. unpow1 76.1%

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

       2. associate-*r* 71.5%

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

       3. *-commutative 71.5%

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

       4. associate-/r/ 76.3%

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

       5. associate-*r/ 71.5%

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

       6. associate-*l/ 76.0%

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

       7. associate-/r/ 76.3%

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

       8. associate-*l/ 82.4%

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

       9. associate-*r/ 79.2%

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

       10. associate-*l/ 74.4%

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

       11. associate-*r* 74.4%

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

       12. *-commutative 74.4%

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

       13. associate-*l* 74.4%

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

       14. associate-*r* 74.4%

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

       15. *-commutative 74.4%

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

       16. associate-*r* 76.2%

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

       17. associate-*r/ 76.2%

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

   12. Simplified 79.1%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{0.25}{\frac{d}{D \cdot M}} \cdot \left(D \cdot \left(M \cdot \frac{h}{d}\right)\right)}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{\frac{d}{M \cdot D}} \cdot \left(D \cdot \left(M \cdot \frac{h}{d}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{-203}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\frac{d}{M \cdot D}} \cdot \left(\left(M \cdot \frac{h}{d}\right) \cdot \frac{D}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 6e-203)
   w0
   (* w0 (sqrt (- 1.0 (* (/ 0.25 (/ d (* M D))) (* (* M (/ h d)) (/ D l))))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6e-203) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 / (d / (M * D))) * ((M * (h / d)) * (D / l)))));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 6d-203) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / (d_1 / (m * d))) * ((m * (h / d_1)) * (d / l)))))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6e-203) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / (d / (M * D))) * ((M * (h / d)) * (D / l)))));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6e-203:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / (d / (M * D))) * ((M * (h / d)) * (D / l)))))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 6e-203)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / Float64(d / Float64(M * D))) * Float64(Float64(M * Float64(h / d)) * Float64(D / l))))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 6e-203)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((0.25 / (d / (M * D))) * ((M * (h / d)) * (D / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 6e-203], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.0000000000000002e-203

   1. Initial program 85.7%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in D around 0 74.5%

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

   if 6.0000000000000002e-203 < M

   1. Initial program 78.4%

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

   3. Simplified 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l/ 78.4%

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

      3. associate-*r/ 79.2%

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

      4. times-frac 78.4%

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

      5. unpow2 78.4%

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

      6. clear-num 78.4%

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

      7. clear-num 78.4%

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

      8. frac-times 78.4%

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

      9. metadata-eval 78.4%

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

      10. times-frac 78.4%

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

      11. times-frac 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-/r* 79.2%

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

      2. associate-/r* 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 87.2%

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

      2. associate-/r/ 86.3%

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

      3. associate-/r/ 86.3%

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

      4. metadata-eval 86.3%

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

   10. Applied egg-rr 86.3%

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

       1. associate-*r/ 86.3%

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

   12. Applied egg-rr 86.3%

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

       1. associate-*r/ 86.3%

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

       2. frac-times 85.5%

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

       3. associate-/l* 85.5%

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

       4. *-commutative 85.5%

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

       5. associate-*r/ 85.5%

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

       6. associate-*r* 85.5%

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

       7. frac-times 86.3%

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

       8. associate-*l* 85.4%

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

   14. Applied egg-rr 85.4%

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

       1. associate-*r/ 84.5%

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

       2. associate-*r/ 84.5%

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

       3. associate-*l/ 84.5%

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

       4. associate-*r* 84.5%

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

       5. *-commutative 84.5%

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

       6. associate-*l* 84.5%

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

       7. associate-*r* 84.5%

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

       8. *-commutative 84.5%

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

       9. associate-*r* 83.7%

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

       10. associate-*r/ 83.7%

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

       11. associate-*r* 85.5%

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

       12. *-commutative 85.5%

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

       13. associate-*l/ 85.2%

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

       14. associate-/l* 85.2%

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

       15. times-frac 84.4%

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

       16. associate-/l/ 82.0%

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

   16. Simplified 83.6%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\frac{d}{D \cdot M}} \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{h}{d}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{-203}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\frac{d}{M \cdot D}} \cdot \left(\left(M \cdot \frac{h}{d}\right) \cdot \frac{D}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 - (((((0.5d0 * m) / d_1) * d) * h) / (l * ((2.0d0 / m) * (d_1 / d))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 82.3%

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

   1. *-commutative 82.3%

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

   2. associate-*l/ 82.5%

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

   3. associate-*r/ 81.7%

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

   4. times-frac 82.5%

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

   5. unpow2 82.5%

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

   6. clear-num 82.5%

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

   7. clear-num 82.5%

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

   8. frac-times 82.5%

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

   9. metadata-eval 82.5%

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

   10. times-frac 81.3%

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

   11. times-frac 82.1%

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

6. Applied egg-rr 82.1%

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

   1. associate-/r* 82.1%

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

   2. associate-/r* 82.1%

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

8. Simplified 82.1%

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

   1. frac-times 90.1%

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

   2. associate-/r/ 89.6%

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

   3. associate-/r/ 89.6%

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

   4. metadata-eval 89.6%

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

10. Applied egg-rr 89.6%

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

11. Final simplification 89.6%

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

Alternative 5: 83.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 - ((0.5d0 * (d * ((m * h) / d_1))) / (l * ((2.0d0 / m) * (d_1 / d))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 82.3%

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

   1. *-commutative 82.3%

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

   2. associate-*l/ 82.5%

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

   3. associate-*r/ 81.7%

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

   4. times-frac 82.5%

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

   5. unpow2 82.5%

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

   6. clear-num 82.5%

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

   7. clear-num 82.5%

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

   8. frac-times 82.5%

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

   9. metadata-eval 82.5%

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

   10. times-frac 81.3%

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

   11. times-frac 82.1%

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

6. Applied egg-rr 82.1%

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

   1. associate-/r* 82.1%

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

   2. associate-/r* 82.1%

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

8. Simplified 82.1%

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

   1. frac-times 90.1%

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

   2. associate-/r/ 89.6%

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

   3. associate-/r/ 89.6%

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

   4. metadata-eval 89.6%

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

10. Applied egg-rr 89.6%

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

11. Taylor expanded in M around 0 86.1%

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

    1. associate-/l* 86.5%

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

13. Simplified 86.5%

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

14. Final simplification 86.5%

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

Alternative 6: 67.1% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 82.5%

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

3. Simplified 82.3%

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

5. Taylor expanded in D around 0 68.7%

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

Reproduce

herbie shell --seed 2024151 
(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))))))