Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 87.3%

Time: 21.1s

Alternatives: 11

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: 80.5% 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: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+172}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(M \cdot h\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 5e+172)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (- 1.0 (* (/ 0.25 l) (* (/ D d) (* M (* (/ D d) (* M h)))))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+172) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
	}
	return tmp;
}

Fortran

NOTE: M 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+172) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / l) * ((d / d_1) * (m * ((d / d_1) * (m * h)))))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+172) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+172:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+172)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(D / d) * Float64(M * Float64(Float64(D / d) * Float64(M * h))))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+172)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e+172], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M * N[(N[(D / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.0000000000000001e172

   1. Initial program 99.9%

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

   if 5.0000000000000001e172 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 42.8%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in D around 0 42.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      2. *-commutative 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}}} \]

      3. times-frac 44.9%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}} \]

      4. *-commutative 44.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{{d}^{2}}} \]

      5. associate-/l* 44.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\frac{{d}^{2}}{{D}^{2}}}}} \]

      6. unpow2 44.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\frac{{d}^{2}}{{D}^{2}}}} \]

      7. associate-*l* 47.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\frac{{d}^{2}}{{D}^{2}}}} \]

      8. unpow2 47.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}} \]

      9. unpow2 47.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}} \]

   6. Simplified 47.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 47.3%

         \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}}}\right)} \]

      2. *-commutative 47.3%

         \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}} \cdot \frac{0.25}{\ell}}}\right) \]

      3. times-frac 59.7%

         \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \cdot \frac{0.25}{\ell}}\right) \]

   8. Applied egg-rr 59.7%

      \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{0.25}{\ell}}\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 59.7%

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{0.25}{\ell}}} \]

      2. associate-*l/ 54.8%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{D} \cdot \frac{d}{D}}}} \]

      3. times-frac 58.6%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}} \cdot \frac{\frac{0.25}{\ell}}{\frac{d}{D}}}} \]

      4. *-commutative 58.6%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}} \]

   10. Simplified 58.6%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 58.4%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)\right)} \]

       2. expm1-udef 58.4%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right)} \]

       3. div-inv 58.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\left(\frac{0.25}{\ell} \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right) \]

       4. clear-num 58.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \color{blue}{\frac{D}{d}}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right) \]

       5. div-inv 58.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{1}{\frac{d}{D}}\right)}}\right)} - 1\right) \]

       6. clear-num 58.4%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \color{blue}{\frac{D}{d}}\right)}\right)} - 1\right) \]

   12. Applied egg-rr 58.4%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 58.4%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}\right)\right)} \]

       2. expm1-log1p 58.6%

          \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}} \]

       3. associate-*l* 60.9%

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)\right)}} \]

       4. associate-*l* 62.2%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{D}{d}\right)\right)}\right)} \]

   14. Simplified 62.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+172}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(M \cdot h\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -4e+181)
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M))))))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e+181) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M 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 ((h / l) <= (-4d+181)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d / d_1) * ((d / d_1) * ((m * h) / (l / m)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / d_1)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e+181) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -4e+181:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -4e+181)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -4e+181)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -4e+181], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -3.9999999999999997e181

   1. Initial program 53.2%

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

   3. Simplified 51.0%

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

      1. *-commutative 51.0%

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

      2. frac-times 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*l/ 73.3%

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

      5. div-inv 73.3%

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

      6. associate-*l* 73.3%

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

      7. associate-/r* 73.3%

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

      8. metadata-eval 73.3%

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

   5. Applied egg-rr 73.3%

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

   6. Taylor expanded in h around 0 45.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow2 45.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      2. times-frac 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}} \]

      3. unpow2 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)} \]

      4. *-commutative 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{\ell}\right)} \]

      5. associate-/r* 48.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\frac{\frac{{D}^{2}}{d}}{d}} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      6. unpow2 48.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      7. associate-*r/ 58.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      8. associate-*l/ 61.0%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      9. *-commutative 61.0%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{\ell}\right)} \]

      10. associate-/l* 50.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}}\right)} \]

      11. associate-*l* 50.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)}} \]

      12. associate-/r/ 61.3%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right)} \]

      13. associate-/l* 65.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\color{blue}{\frac{M}{\frac{\ell}{M}}} \cdot h\right)\right)\right)} \]

      14. associate-*l/ 65.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}}\right)\right)} \]

   8. Simplified 65.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}} \]

   if -3.9999999999999997e181 < (/.f64 h l)

   1. Initial program 87.0%

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

   3. Simplified 87.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \end{array} \]

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -4e+181)
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M))))))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* D 0.5) (/ d M)) 2.0)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e+181) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D * 0.5) / (d / M)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M 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 ((h / l) <= (-4d+181)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d / d_1) * ((d / d_1) * ((m * h) / (l / m)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * 0.5d0) / (d_1 / m)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e+181) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D * 0.5) / (d / M)), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -4e+181:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D * 0.5) / (d / M)), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -4e+181)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D * 0.5) / Float64(d / M)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -4e+181)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * 0.5) / (d / M)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -4e+181], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -3.9999999999999997e181

   1. Initial program 53.2%

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

   3. Simplified 51.0%

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

      1. *-commutative 51.0%

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

      2. frac-times 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*l/ 73.3%

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

      5. div-inv 73.3%

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

      6. associate-*l* 73.3%

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

      7. associate-/r* 73.3%

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

      8. metadata-eval 73.3%

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

   5. Applied egg-rr 73.3%

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

   6. Taylor expanded in h around 0 45.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   7. Step-by-step derivation

      1. unpow2 45.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      2. times-frac 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}} \]

      3. unpow2 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)} \]

      4. *-commutative 45.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{\ell}\right)} \]

      5. associate-/r* 48.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\frac{\frac{{D}^{2}}{d}}{d}} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      6. unpow2 48.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      7. associate-*r/ 58.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      8. associate-*l/ 61.0%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

      9. *-commutative 61.0%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{\ell}\right)} \]

      10. associate-/l* 50.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}}\right)} \]

      11. associate-*l* 50.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)}} \]

      12. associate-/r/ 61.3%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right)} \]

      13. associate-/l* 65.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\color{blue}{\frac{M}{\frac{\ell}{M}}} \cdot h\right)\right)\right)} \]

      14. associate-*l/ 65.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}}\right)\right)} \]

   8. Simplified 65.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}} \]

   if -3.9999999999999997e181 < (/.f64 h l)

   1. Initial program 87.0%

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

   3. Simplified 87.9%

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

      1. clear-num 87.9%

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

      2. un-div-inv 87.9%

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

      3. div-inv 87.9%

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

      4. metadata-eval 87.9%

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

   5. Applied egg-rr 87.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+181}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}\\ \end{array} \]

Alternative 4: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)} \end{array} \]

FPCore

NOTE: M 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 (* 0.25 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
}

Fortran

NOTE: M 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 - (0.25d0 * ((d / d_1) * ((d / d_1) * ((m * h) / (l / m)))))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M))))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (0.25 * ((D / d) * ((D / d) * ((M * h) / (l / M)))))));
end

Wolfram

NOTE: M 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[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

   1. *-commutative 82.3%

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

   2. frac-times 81.9%

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

   3. *-commutative 81.9%

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

   4. associate-*l/ 85.4%

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

   5. div-inv 85.4%

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

   6. associate-*l* 85.4%

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

   7. associate-/r* 85.4%

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

   8. metadata-eval 85.4%

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

5. Applied egg-rr 85.4%

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

6. Taylor expanded in h around 0 54.1%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
7. Step-by-step derivation

   1. unpow2 54.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   2. times-frac 54.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}} \]

   3. unpow2 54.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)} \]

   4. *-commutative 54.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{h \cdot \left(M \cdot M\right)}}{\ell}\right)} \]

   5. associate-/r* 60.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\frac{\frac{{D}^{2}}{d}}{d}} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

   6. unpow2 60.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

   7. associate-*r/ 66.3%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot \frac{D}{d}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

   8. associate-*l/ 67.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)} \]

   9. *-commutative 67.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{\ell}\right)} \]

   10. associate-/l* 66.2%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}}\right)} \]

   11. associate-*l* 71.0%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)}} \]

   12. associate-/r/ 71.6%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right)} \]

   13. associate-/l* 74.6%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\color{blue}{\frac{M}{\frac{\ell}{M}}} \cdot h\right)\right)\right)} \]

   14. associate-*l/ 77.9%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}}\right)\right)} \]

8. Simplified 77.9%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)}} \]

9. Final simplification 77.9%

   \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)} \]

Alternative 5: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(M \cdot h\right)\right)\right)\right)} \end{array} \]

FPCore

NOTE: M 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 (* (/ 0.25 l) (* (/ D d) (* M (* (/ D d) (* M h)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
}

Fortran

NOTE: M 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 - ((0.25d0 / l) * ((d / d_1) * (m * ((d / d_1) * (m * h)))))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(D / d) * Float64(M * Float64(Float64(D / d) * Float64(M * h))))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((D / d) * (M * ((D / d) * (M * h)))))));
end

Wolfram

NOTE: M 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[(0.25 / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M * N[(N[(D / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 54.1%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
5. Step-by-step derivation

   1. associate-*r/ 54.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

   2. *-commutative 54.1%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}}} \]

   3. times-frac 56.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}} \]

   4. *-commutative 56.1%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{{d}^{2}}} \]

   5. associate-/l* 56.9%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\frac{{d}^{2}}{{D}^{2}}}}} \]

   6. unpow2 56.9%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\frac{{d}^{2}}{{D}^{2}}}} \]

   7. associate-*l* 60.0%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\frac{{d}^{2}}{{D}^{2}}}} \]

   8. unpow2 60.0%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}} \]

   9. unpow2 60.0%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}} \]

6. Simplified 60.0%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 60.0%

      \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}}}\right)} \]

   2. *-commutative 60.0%

      \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d \cdot d}{D \cdot D}} \cdot \frac{0.25}{\ell}}}\right) \]

   3. times-frac 73.7%

      \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \cdot \frac{0.25}{\ell}}\right) \]

8. Applied egg-rr 73.7%

   \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{0.25}{\ell}}\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 73.7%

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{0.25}{\ell}}} \]

   2. associate-*l/ 70.6%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{0.25}{\ell}}{\frac{d}{D} \cdot \frac{d}{D}}}} \]

   3. times-frac 76.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}} \cdot \frac{\frac{0.25}{\ell}}{\frac{d}{D}}}} \]

   4. *-commutative 76.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}} \]

10. Simplified 76.4%

    \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 76.3%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)\right)} \]

    2. expm1-udef 76.3%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{0.25}{\ell}}{\frac{d}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right)} \]

    3. div-inv 75.9%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\left(\frac{0.25}{\ell} \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right) \]

    4. clear-num 75.9%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \color{blue}{\frac{D}{d}}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{d}{D}}}\right)} - 1\right) \]

    5. div-inv 75.9%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{1}{\frac{d}{D}}\right)}}\right)} - 1\right) \]

    6. clear-num 75.9%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \color{blue}{\frac{D}{d}}\right)}\right)} - 1\right) \]

12. Applied egg-rr 75.9%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 75.9%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}\right)\right)} \]

    2. expm1-log1p 76.0%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \left(\frac{0.25}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)}} \]

    3. associate-*l* 78.1%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{D}{d}\right)\right)}} \]

    4. associate-*l* 83.0%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{D}{d}\right)\right)}\right)} \]

14. Simplified 83.0%

    \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{D}{d}\right)\right)\right)}} \]

15. Final simplification 83.0%

    \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(\frac{D}{d} \cdot \left(M \cdot h\right)\right)\right)\right)} \]

Alternative 6: 68.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h \cdot w0}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+101)
   w0
   (* -0.125 (* (/ (* D D) (* d d)) (/ (* M M) (/ l (* h w0)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+101) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((M * M) / (l / (h * w0))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 4d+101) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * ((m * m) / (l / (h * w0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+101) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((M * M) / (l / (h * w0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+101:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / (d * d)) * ((M * M) / (l / (h * w0))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+101)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(M * M) / Float64(l / Float64(h * w0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+101)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / (d * d)) * ((M * M) / (l / (h * w0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+101], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.9999999999999999e101

   1. Initial program 84.6%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in D around 0 71.1%

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

   if 3.9999999999999999e101 < M

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in D around 0 32.8%

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

      1. *-commutative 32.8%

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

      2. times-frac 26.1%

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

      3. unpow2 26.1%

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

      4. unpow2 26.1%

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

      5. unpow2 26.1%

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

   6. Simplified 26.1%

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

      1. associate-*r/ 33.0%

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

      2. times-frac 33.2%

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

      3. *-commutative 33.2%

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

   8. Applied egg-rr 33.2%

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

   9. Taylor expanded in h around 0 33.2%

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

       1. unpow2 33.2%

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

       2. associate-*r* 36.7%

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

   11. Simplified 36.7%

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

   12. Taylor expanded in D around inf 19.4%

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

       1. times-frac 19.6%

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

       2. unpow2 19.6%

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

       3. unpow2 19.6%

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

       4. associate-/l* 19.3%

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

       5. unpow2 19.3%

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

   14. Simplified 19.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M \cdot M}{\frac{\ell}{h \cdot w0}}\right)\\ \end{array} \]

Alternative 7: 75.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\left(h \cdot \frac{M}{\ell}\right) \cdot \frac{M \cdot D}{d}\right)\right)\right) \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (* h (/ M l)) (/ (* M D) d)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((h * (M / l)) * ((M * D) / d)))));
}

Fortran

NOTE: M 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 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((h * (m / l)) * ((m * d) / d_1)))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((h * (M / l)) * ((M * D) / d)))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * ((D / d) * ((h * (M / l)) * ((M * D) / d)))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(h * Float64(M / l)) * Float64(Float64(M * D) / d))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((h * (M / l)) * ((M * D) / d)))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(h * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. unpow2 53.1%

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

   5. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in D around 0 52.7%

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

   1. unpow2 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. *-commutative 53.1%

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

   5. associate-/r* 58.7%

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

   6. unpow2 58.7%

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

   7. associate-*r/ 63.8%

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

   8. associate-*l/ 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-/l* 64.1%

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

   11. associate-*l* 68.6%

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

   12. associate-/r/ 68.7%

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

   13. associate-/l* 70.6%

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

   14. associate-*l/ 72.9%

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

9. Simplified 72.9%

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

10. Taylor expanded in D around 0 67.4%

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

    1. times-frac 69.1%

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

    2. unpow2 69.1%

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

    3. associate-*r* 71.0%

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

    4. *-commutative 71.0%

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

    5. associate-*l/ 72.9%

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

    6. *-commutative 72.9%

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

    7. associate-*l* 74.5%

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

    8. *-commutative 74.5%

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

    9. associate-*l/ 73.8%

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

12. Simplified 73.8%

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

    1. expm1-log1p-u 65.6%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{M \cdot h}{\ell}\right)\right)} \cdot \frac{D \cdot M}{d}\right)\right) \cdot -0.125\right) \]

    2. expm1-udef 63.5%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{M \cdot h}{\ell}\right)} - 1\right)} \cdot \frac{D \cdot M}{d}\right)\right) \cdot -0.125\right) \]

    3. associate-/l* 64.6%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{M}{\frac{\ell}{h}}}\right)} - 1\right) \cdot \frac{D \cdot M}{d}\right)\right) \cdot -0.125\right) \]

14. Applied egg-rr 64.6%

    \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{M}{\frac{\ell}{h}}\right)} - 1\right)} \cdot \frac{D \cdot M}{d}\right)\right) \cdot -0.125\right) \]
15. Step-by-step derivation

    1. expm1-def 67.0%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{M}{\frac{\ell}{h}}\right)\right)} \cdot \frac{D \cdot M}{d}\right)\right) \cdot -0.125\right) \]

    2. expm1-log1p 74.8%

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

    3. associate-/r/ 74.9%

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

16. Simplified 74.9%

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

17. Final simplification 74.9%

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

Alternative 8: 73.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (* M (* h (/ M l)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (M * (h * (M / l)))))));
}

Fortran

NOTE: M 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 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((d / d_1) * (m * (h * (m / l)))))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (M * (h * (M / l)))))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (M * (h * (M / l)))))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(M * Float64(h * Float64(M / l))))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (M * (h * (M / l)))))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M * N[(h * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. unpow2 53.1%

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

   5. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in D around 0 52.7%

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

   1. unpow2 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. *-commutative 53.1%

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

   5. associate-/r* 58.7%

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

   6. unpow2 58.7%

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

   7. associate-*r/ 63.8%

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

   8. associate-*l/ 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-/l* 64.1%

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

   11. associate-*l* 68.6%

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

   12. associate-/r/ 68.7%

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

   13. associate-/l* 70.6%

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

   14. associate-*l/ 72.9%

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

9. Simplified 72.9%

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

10. Taylor expanded in D around 0 67.4%

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

    1. times-frac 69.1%

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

    2. unpow2 69.1%

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

    3. associate-*r* 71.0%

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

    4. *-commutative 71.0%

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

    5. associate-*l/ 72.9%

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

    6. *-commutative 72.9%

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

    7. associate-*l* 74.5%

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

    8. *-commutative 74.5%

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

    9. associate-*l/ 73.8%

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

12. Simplified 73.8%

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

13. Taylor expanded in M around 0 67.4%

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

    1. times-frac 69.1%

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

    2. associate-/l* 68.6%

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

    3. unpow2 68.6%

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

    4. associate-*r/ 72.1%

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

    5. associate-/r/ 73.3%

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

    6. *-commutative 73.3%

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

15. Simplified 73.3%

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

16. Final simplification 73.3%

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

Alternative 9: 75.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ (* M h) l) (* M (/ D d))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * (((M * h) / l) * (M * (D / d))))));
}

Fortran

NOTE: M 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 * (1.0d0 + ((-0.125d0) * ((d / d_1) * (((m * h) / l) * (m * (d / d_1))))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * (((M * h) / l) * (M * (D / d))))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * ((D / d) * (((M * h) / l) * (M * (D / d))))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(Float64(M * h) / l) * Float64(M * Float64(D / d)))))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((D / d) * (((M * h) / l) * (M * (D / d))))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision] * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. unpow2 53.1%

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

   5. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in D around 0 52.7%

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

   1. unpow2 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. *-commutative 53.1%

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

   5. associate-/r* 58.7%

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

   6. unpow2 58.7%

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

   7. associate-*r/ 63.8%

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

   8. associate-*l/ 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-/l* 64.1%

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

   11. associate-*l* 68.6%

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

   12. associate-/r/ 68.7%

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

   13. associate-/l* 70.6%

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

   14. associate-*l/ 72.9%

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

9. Simplified 72.9%

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

10. Taylor expanded in D around 0 67.4%

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

    1. times-frac 69.1%

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

    2. unpow2 69.1%

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

    3. associate-*r* 71.0%

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

    4. *-commutative 71.0%

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

    5. associate-*l/ 72.9%

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

    6. *-commutative 72.9%

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

    7. associate-*l* 74.5%

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

    8. *-commutative 74.5%

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

    9. associate-*l/ 73.8%

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

12. Simplified 73.8%

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

    1. expm1-log1p-u 65.4%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot M}{d}\right)\right)}\right)\right) \cdot -0.125\right) \]

    2. expm1-udef 65.2%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot M}{d}\right)} - 1\right)}\right)\right) \cdot -0.125\right) \]

    3. associate-/l* 66.0%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{d}{M}}}\right)} - 1\right)\right)\right) \cdot -0.125\right) \]

14. Applied egg-rr 66.0%

    \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{d}{M}}\right)} - 1\right)}\right)\right) \cdot -0.125\right) \]
15. Step-by-step derivation

    1. expm1-def 66.1%

       \[\leadsto w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \left(\frac{M \cdot h}{\ell} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{d}{M}}\right)\right)}\right)\right) \cdot -0.125\right) \]

    2. expm1-log1p 74.5%

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

    3. associate-/r/ 74.5%

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

16. Simplified 74.5%

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

17. Final simplification 74.5%

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

Alternative 10: 76.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + \left(\frac{D}{d} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot h\right)}{\frac{\ell}{M}}\right) \cdot -0.125\right) \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* (* (/ D d) (/ (* (/ D d) (* M h)) (/ l M))) -0.125))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (((D / d) * (((D / d) * (M * h)) / (l / M))) * -0.125));
}

Fortran

NOTE: M 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 * (1.0d0 + (((d / d_1) * (((d / d_1) * (m * h)) / (l / m))) * (-0.125d0)))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (((D / d) * (((D / d) * (M * h)) / (l / M))) * -0.125));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (((D / d) * (((D / d) * (M * h)) / (l / M))) * -0.125))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D / d) * Float64(Float64(Float64(D / d) * Float64(M * h)) / Float64(l / M))) * -0.125)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (((D / d) * (((D / d) * (M * h)) / (l / M))) * -0.125));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. unpow2 53.1%

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

   5. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in D around 0 52.7%

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

   1. unpow2 52.7%

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

   2. times-frac 53.1%

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

   3. unpow2 53.1%

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

   4. *-commutative 53.1%

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

   5. associate-/r* 58.7%

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

   6. unpow2 58.7%

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

   7. associate-*r/ 63.8%

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

   8. associate-*l/ 64.6%

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

   9. *-commutative 64.6%

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

   10. associate-/l* 64.1%

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

   11. associate-*l* 68.6%

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

   12. associate-/r/ 68.7%

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

   13. associate-/l* 70.6%

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

   14. associate-*l/ 72.9%

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

9. Simplified 72.9%

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

    1. associate-*r/ 75.4%

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

11. Applied egg-rr 75.4%

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

12. Final simplification 75.4%

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

Alternative 11: 67.8% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 82.3%

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

4. Taylor expanded in D around 0 68.0%

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

5. Final simplification 68.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023287 
(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))))))