Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 86.4%

Time: 17.0s

Alternatives: 12

Speedup: 1.7×

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: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{M}{\frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l)) 1e-9)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (/ M d) (/ 2.0 D)) 2.0)))))
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ M (/ d (* h M))) (/ (* D D) d))) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= 1d-9) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) / (2.0d0 / d)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((m / (d_1 / (h * m))) * ((d * d) / d_1))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) / (2.0 / D)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= 1e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) / Float64(2.0 / D)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M / Float64(d / Float64(h * M))) * Float64(Float64(D * D) / d))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((D * M) / (d * 2.0)) ^ 2.0) * (h / l)) <= 1e-9)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) / (2.0 / D)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-9], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] / N[(2.0 / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(M / N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 1.00000000000000006e-9

   1. Initial program 86.4%

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

      1. times-frac 86.1%

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

   3. Simplified 86.1%

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

      1. associate-*l/ 86.1%

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

      2. associate-*r/ 86.4%

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

      3. associate-*l/ 86.5%

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

      4. associate-/l* 86.5%

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

   5. Applied egg-rr 86.5%

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

   if 1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. frac-times 0.0%

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

      4. associate-*r/ 64.4%

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

      5. frac-times 64.4%

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

      6. unpow2 64.4%

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

      7. unpow2 64.4%

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

      8. frac-times 64.4%

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

      9. div-inv 64.4%

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

      10. associate-*l* 64.4%

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

      11. associate-/r* 64.4%

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

      12. metadata-eval 64.4%

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

   5. Applied egg-rr 64.4%

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

   6. Taylor expanded in M around 0 57.3%

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

      1. unpow2 57.3%

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

      2. *-commutative 57.3%

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

      3. *-commutative 57.3%

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

      4. unpow2 57.3%

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

      5. associate-*l* 57.3%

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

      6. times-frac 75.7%

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

      7. *-commutative 75.7%

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

      8. associate-/l* 75.7%

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

      9. unpow2 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{M}{\frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{d}\right)}{\ell}}\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{M}{\frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l)) 1e-9)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ M (/ d (* h M))) (/ (* D D) d))) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= 1d-9) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((m / (d_1 / (h * m))) * ((d * d) / d_1))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l)) <= 1e-9:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= 1e-9)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M / Float64(d / Float64(h * M))) * Float64(Float64(D * D) / d))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((D * M) / (d * 2.0)) ^ 2.0) * (h / l)) <= 1e-9)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M / (d / (h * M))) * ((D * D) / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-9], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(M / N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 1.00000000000000006e-9

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. times-frac 86.5%

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

   3. Simplified 86.5%

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

   if 1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. frac-times 0.0%

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

      4. associate-*r/ 64.4%

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

      5. frac-times 64.4%

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

      6. unpow2 64.4%

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

      7. unpow2 64.4%

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

      8. frac-times 64.4%

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

      9. div-inv 64.4%

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

      10. associate-*l* 64.4%

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

      11. associate-/r* 64.4%

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

      12. metadata-eval 64.4%

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

   5. Applied egg-rr 64.4%

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

   6. Taylor expanded in M around 0 57.3%

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

      1. unpow2 57.3%

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

      2. *-commutative 57.3%

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

      3. *-commutative 57.3%

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

      4. unpow2 57.3%

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

      5. associate-*l* 57.3%

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

      6. times-frac 75.7%

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

      7. *-commutative 75.7%

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

      8. associate-/l* 75.7%

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

      9. unpow2 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{M}{\frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{d}\right)}{\ell}}\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 4e+113)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M)))) l))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4e+113) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d <= 4d+113) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / 2.0d0) * (d / d_1)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4e+113) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 4e+113:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4e+113)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M)))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 4e+113)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4e+113], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4e113

   1. Initial program 81.3%

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

      1. times-frac 81.3%

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

   3. Simplified 81.3%

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

      1. unpow2 81.3%

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

      2. unpow2 81.3%

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

      3. frac-times 81.3%

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

      4. associate-*r/ 85.9%

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

      5. frac-times 86.0%

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

      6. unpow2 86.0%

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

      7. unpow2 86.0%

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

      8. frac-times 85.9%

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

      9. div-inv 85.9%

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

      10. associate-*l* 86.0%

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

      11. associate-/r* 86.0%

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

      12. metadata-eval 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in M around 0 56.7%

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

      1. associate-/l* 56.7%

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

      2. unpow2 56.7%

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

      3. *-commutative 56.7%

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

      4. unpow2 56.7%

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

      5. associate-*l* 59.0%

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

      6. unpow2 59.0%

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

      7. times-frac 69.1%

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

   8. Simplified 69.1%

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

   if 4e113 < D

   1. Initial program 79.6%

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

      1. times-frac 77.0%

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

   3. Simplified 77.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \end{array} \]

Alternative 4: 86.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (/ (* h (pow (* D (* M (/ 0.5 d))) 2.0)) l)))))

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 - ((h * ((d * (m * (0.5d0 / d_1))) ** 2.0d0)) / l)))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 - ((h * Math.pow((D * (M * (0.5 / d))), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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[(h * N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. times-frac 80.7%

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

3. Simplified 80.7%

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

   1. associate-*r/ 84.4%

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

   2. clear-num 84.4%

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

   3. frac-times 84.7%

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

   4. div-inv 84.7%

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

   5. associate-*l* 84.4%

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

   6. associate-/r* 84.4%

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

   7. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

   1. associate-/r/ 84.3%

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

   2. *-commutative 84.3%

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

   3. associate-*r* 84.6%

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

   4. *-commutative 84.6%

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

   5. associate-*r* 84.7%

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

7. Simplified 84.7%

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

   1. associate-*l/ 84.7%

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

   2. *-un-lft-identity 84.7%

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

9. Applied egg-rr 84.7%

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

10. Final simplification 84.7%

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

Alternative 5: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 3 \cdot 10^{+80}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{0.25}{\frac{\ell}{D \cdot \left(D \cdot \left(M \cdot \frac{h \cdot M}{d \cdot d}\right)\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 3e+80)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M)))) l))))
   (*
    w0
    (pow (- 1.0 (/ 0.25 (/ l (* D (* D (* M (/ (* h M) (* d d)))))))) 0.5))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3e+80) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * pow((1.0 - (0.25 / (l / (D * (D * (M * ((h * M) / (d * d)))))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d <= 3d+80) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))) / l)))
    else
        tmp = w0 * ((1.0d0 - (0.25d0 / (l / (d * (d * (m * ((h * m) / (d_1 * d_1)))))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3e+80) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * Math.pow((1.0 - (0.25 / (l / (D * (D * (M * ((h * M) / (d * d)))))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 3e+80:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)))
	else:
		tmp = w0 * math.pow((1.0 - (0.25 / (l / (D * (D * (M * ((h * M) / (d * d)))))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 3e+80)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M)))) / l))));
	else
		tmp = Float64(w0 * (Float64(1.0 - Float64(0.25 / Float64(l / Float64(D * Float64(D * Float64(M * Float64(Float64(h * M) / Float64(d * d)))))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 3e+80)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	else
		tmp = w0 * ((1.0 - (0.25 / (l / (D * (D * (M * ((h * M) / (d * d)))))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 3e+80], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 - N[(0.25 / N[(l / N[(D * N[(D * N[(M * N[(N[(h * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.99999999999999987e80

   1. Initial program 81.4%

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

      1. times-frac 81.1%

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

   3. Simplified 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. frac-times 81.4%

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

      4. associate-*r/ 86.2%

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

      5. frac-times 85.8%

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

      6. unpow2 85.8%

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

      7. unpow2 85.8%

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

      8. frac-times 86.2%

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

      9. div-inv 86.2%

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

      10. associate-*l* 85.8%

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

      11. associate-/r* 85.8%

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

      12. metadata-eval 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in M around 0 56.6%

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

      1. associate-/l* 56.6%

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

      2. unpow2 56.6%

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

      3. *-commutative 56.6%

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

      4. unpow2 56.6%

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

      5. associate-*l* 58.9%

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

      6. unpow2 58.9%

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

      7. times-frac 68.7%

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

   8. Simplified 68.7%

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

   if 2.99999999999999987e80 < D

   1. Initial program 78.7%

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

      1. times-frac 78.7%

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

   3. Simplified 78.7%

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

      1. associate-*r/ 76.5%

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

      2. clear-num 76.5%

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

      3. frac-times 76.4%

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

      4. div-inv 76.4%

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

      5. associate-*l* 76.5%

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

      6. associate-/r* 76.5%

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

      7. metadata-eval 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. associate-/r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r* 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*r* 78.8%

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

   7. Simplified 78.8%

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

      1. associate-*l/ 78.8%

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

      2. *-un-lft-identity 78.8%

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

   9. Applied egg-rr 78.8%

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

   10. Taylor expanded in h around 0 33.3%

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

       1. unpow2 33.3%

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

       2. *-commutative 33.3%

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

       3. associate-*r* 35.9%

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

       4. unpow2 35.9%

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

       5. times-frac 33.7%

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

       6. associate-*r/ 36.2%

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

       7. associate-*r* 33.9%

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

       8. unpow2 33.9%

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

       9. *-commutative 33.9%

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

       10. times-frac 33.5%

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

       11. unpow2 33.5%

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

       12. associate-*r* 33.5%

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

       13. associate-/l* 38.9%

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

       14. unpow2 38.9%

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

   12. Simplified 38.9%

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

       1. pow1/2 39.0%

          \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \frac{h \cdot M}{\frac{d \cdot d}{M}}\right)}{\ell}\right)}^{0.5}} \]

       2. associate-/l* 39.0%

          \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{0.25}{\frac{\ell}{\left(D \cdot D\right) \cdot \frac{h \cdot M}{\frac{d \cdot d}{M}}}}}\right)}^{0.5} \]

       3. associate-*l* 69.4%

          \[\leadsto w0 \cdot {\left(1 - \frac{0.25}{\frac{\ell}{\color{blue}{D \cdot \left(D \cdot \frac{h \cdot M}{\frac{d \cdot d}{M}}\right)}}}\right)}^{0.5} \]

       4. associate-/r/ 69.4%

          \[\leadsto w0 \cdot {\left(1 - \frac{0.25}{\frac{\ell}{D \cdot \left(D \cdot \color{blue}{\left(\frac{h \cdot M}{d \cdot d} \cdot M\right)}\right)}}\right)}^{0.5} \]

   14. Applied egg-rr 69.4%

       \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{0.25}{\frac{\ell}{D \cdot \left(D \cdot \left(\frac{h \cdot M}{d \cdot d} \cdot M\right)\right)}}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3 \cdot 10^{+80}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{0.25}{\frac{\ell}{D \cdot \left(D \cdot \left(M \cdot \frac{h \cdot M}{d \cdot d}\right)\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 6: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{+148}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{D}{\ell} \cdot \left(M \cdot \frac{h \cdot M}{d}\right)}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.2e+148)
   (* w0 (+ 1.0 (* -0.125 (/ (* (/ D l) (* M (/ (* h M) d))) (/ d D)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.2e+148) {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d_1 <= 1.2d+148) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / l) * (m * ((h * m) / d_1))) / (d_1 / d))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.2e+148) {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.2e+148:
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.2e+148)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / l) * Float64(M * Float64(Float64(h * M) / d))) / Float64(d / D)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.2e+148)
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.2e+148], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.19999999999999997e148

   1. Initial program 80.2%

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

      1. times-frac 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. *-commutative 48.6%

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

      5. times-frac 50.4%

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

      6. unpow2 50.4%

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

      7. *-commutative 50.4%

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

      8. unpow2 50.4%

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

      9. unpow2 50.4%

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

   6. Simplified 50.4%

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

      1. associate-*r/ 50.9%

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

      2. associate-/l* 55.9%

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

      3. *-commutative 55.9%

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

   8. Applied egg-rr 55.9%

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

      1. add-cbrt-cube 55.9%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}}} \cdot -0.125\right) \]

      2. times-frac 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      3. associate-/r/ 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      4. associate-*r* 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      5. times-frac 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      6. associate-/r/ 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      7. associate-*r* 55.4%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      8. times-frac 62.5%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}} \cdot -0.125\right) \]

   10. Applied egg-rr 65.6%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)}} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. associate-*l* 65.6%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)}} \cdot -0.125\right) \]

       2. *-commutative 65.6%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)} \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       3. associate-/l* 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       4. associate-/l* 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       5. associate-*l* 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)}} \cdot -0.125\right) \]

       6. associate-/l* 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       7. associate-/l* 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       8. *-commutative 65.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)}\right)\right)} \cdot -0.125\right) \]

       9. associate-/l* 68.0%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)\right)\right)} \cdot -0.125\right) \]

       10. associate-/l* 71.7%

           \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right)\right)\right)} \cdot -0.125\right) \]

   12. Simplified 71.7%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right)}} \cdot -0.125\right) \]
   13. Step-by-step derivation

       1. *-commutative 71.7%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)}} \cdot -0.125\right) \]

       2. associate-*r* 71.7%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \frac{h \cdot M}{\frac{d}{M}}\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       3. *-commutative 71.7%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       4. add-cbrt-cube 71.8%

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

       5. associate-*r/ 74.2%

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

       6. associate-/r/ 74.2%

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

   14. Applied egg-rr 74.2%

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

   if 1.19999999999999997e148 < d

   1. Initial program 86.1%

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

      1. times-frac 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r/ 91.7%

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

      2. clear-num 91.7%

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

      3. frac-times 89.0%

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

      4. div-inv 89.0%

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

      5. associate-*l* 91.7%

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

      6. associate-/r* 91.7%

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

      7. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. associate-/r/ 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*r* 89.0%

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

      4. *-commutative 89.0%

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

      5. associate-*r* 91.7%

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

   7. Simplified 91.7%

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

      1. associate-*l/ 91.7%

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

      2. *-un-lft-identity 91.7%

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

   9. Applied egg-rr 91.7%

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

   10. Taylor expanded in w0 around 0 51.0%

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

       1. *-commutative 51.0%

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

       2. *-commutative 51.0%

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

       3. *-commutative 51.0%

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

   12. Simplified 71.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{+148}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{D}{\ell} \cdot \left(M \cdot \frac{h \cdot M}{d}\right)}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \]

Alternative 7: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(h \cdot M\right) \cdot \frac{D}{\ell}}{\frac{d}{M} \cdot \frac{d}{D}} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 1.9e+152)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (/ (* D D) (* (/ d (* h M)) (/ d M)))) l))))
   (* w0 (+ 1.0 (* (/ (* (* h M) (/ D l)) (* (/ d M) (/ d D))) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.9e+152) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d <= 1.9d+152) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((d * d) / ((d_1 / (h * m)) * (d_1 / m)))) / l)))
    else
        tmp = w0 * (1.0d0 + ((((h * m) * (d / l)) / ((d_1 / m) * (d_1 / d))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.9e+152) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	} else {
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1.9e+152:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)))
	else:
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1.9e+152)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d / Float64(h * M)) * Float64(d / M)))) / l))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(h * M) * Float64(D / l)) / Float64(Float64(d / M) * Float64(d / D))) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1.9e+152)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D * D) / ((d / (h * M)) * (d / M)))) / l)));
	else
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 1.9e+152], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(h * M), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.9e152

   1. Initial program 80.8%

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

      1. times-frac 80.9%

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

   3. Simplified 80.9%

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

      1. unpow2 80.9%

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

      2. unpow2 80.9%

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

      3. frac-times 80.8%

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

      4. associate-*r/ 85.4%

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

      5. frac-times 85.5%

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

      6. unpow2 85.5%

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

      7. unpow2 85.5%

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

      8. frac-times 85.4%

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

      9. div-inv 85.4%

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

      10. associate-*l* 85.5%

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

      11. associate-/r* 85.5%

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

      12. metadata-eval 85.5%

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

   5. Applied egg-rr 85.5%

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

   6. Taylor expanded in M around 0 56.4%

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

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. *-commutative 56.8%

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

      4. unpow2 56.8%

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

      5. associate-*l* 59.1%

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

      6. unpow2 59.1%

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

      7. times-frac 68.9%

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

   8. Simplified 68.9%

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

   if 1.9e152 < D

   1. Initial program 82.3%

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

      1. times-frac 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in M around 0 25.9%

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

      1. associate-*r/ 25.9%

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

      2. *-commutative 25.9%

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

      3. associate-*r/ 25.9%

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

      4. *-commutative 25.9%

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

      5. times-frac 25.6%

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

      6. unpow2 25.6%

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

      7. *-commutative 25.6%

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

      8. unpow2 25.6%

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

      9. unpow2 25.6%

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

   6. Simplified 25.6%

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

      1. associate-*r/ 25.9%

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

      2. associate-/l* 57.0%

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

      3. *-commutative 57.0%

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

   8. Applied egg-rr 57.0%

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

      1. add-cbrt-cube 57.0%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}}} \cdot -0.125\right) \]

      2. times-frac 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      3. associate-/r/ 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      4. associate-*r* 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      5. times-frac 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      6. associate-/r/ 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      7. associate-*r* 56.9%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      8. times-frac 57.2%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}} \cdot -0.125\right) \]

   10. Applied egg-rr 57.2%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)}} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. associate-*l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)}} \cdot -0.125\right) \]

       2. *-commutative 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)} \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       3. associate-/l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       4. associate-/l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       5. associate-*l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)}} \cdot -0.125\right) \]

       6. associate-/l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       7. associate-/l* 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       8. *-commutative 57.2%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)}\right)\right)} \cdot -0.125\right) \]

       9. associate-/l* 63.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)\right)\right)} \cdot -0.125\right) \]

       10. associate-/l* 66.9%

           \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right)\right)\right)} \cdot -0.125\right) \]

   12. Simplified 66.9%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right)}} \cdot -0.125\right) \]
   13. Step-by-step derivation

       1. *-commutative 66.9%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)}} \cdot -0.125\right) \]

       2. associate-*r* 66.9%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \frac{h \cdot M}{\frac{d}{M}}\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       3. *-commutative 66.9%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       4. add-cbrt-cube 67.0%

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

       5. frac-times 70.6%

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

   14. Applied egg-rr 70.6%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{\left(h \cdot M\right) \cdot \frac{D}{\ell}}{\frac{d}{M} \cdot \frac{d}{D}}} \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{D \cdot D}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(h \cdot M\right) \cdot \frac{D}{\ell}}{\frac{d}{M} \cdot \frac{d}{D}} \cdot -0.125\right)\\ \end{array} \]

Alternative 8: 76.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 4.5e-117)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.5e-117) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 4.5d-117) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.5e-117) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.5e-117:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.5e-117)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.5e-117)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((M * M) / (l / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.5e-117], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.49999999999999969e-117

   1. Initial program 81.7%

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

      1. times-frac 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 68.9%

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

   if 4.49999999999999969e-117 < M

   1. Initial program 79.4%

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

      1. times-frac 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 53.9%

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

      1. associate-*r/ 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-*r/ 53.9%

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

      4. *-commutative 53.9%

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

      5. times-frac 49.9%

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

      6. unpow2 49.9%

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

      7. *-commutative 49.9%

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

      8. unpow2 49.9%

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

      9. unpow2 49.9%

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

   6. Simplified 49.9%

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

      1. associate-*r/ 50.1%

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

      2. associate-/l* 52.6%

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

      3. *-commutative 52.6%

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

   8. Applied egg-rr 52.6%

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

   9. Taylor expanded in D around 0 53.9%

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

       1. times-frac 53.9%

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

       2. *-commutative 53.9%

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

       3. unpow2 53.9%

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

       4. unpow2 53.9%

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

       5. times-frac 65.9%

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

       6. associate-/l* 64.5%

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

       7. unpow2 64.5%

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

   11. Simplified 64.5%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)} \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 9: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(h \cdot M\right) \cdot \frac{D}{\ell}}{\frac{d}{M} \cdot \frac{d}{D}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.02e+120)
   (* w0 (+ 1.0 (* (/ (* (* h M) (/ D l)) (* (/ d M) (/ d D))) -0.125)))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.02e+120) {
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d_1 <= 1.02d+120) then
        tmp = w0 * (1.0d0 + ((((h * m) * (d / l)) / ((d_1 / m) * (d_1 / d))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.02e+120) {
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.02e+120:
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.02e+120)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(h * M) * Float64(D / l)) / Float64(Float64(d / M) * Float64(d / D))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.02e+120)
		tmp = w0 * (1.0 + ((((h * M) * (D / l)) / ((d / M) * (d / D))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.02e+120], N[(w0 * N[(1.0 + N[(N[(N[(N[(h * M), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.01999999999999997e120

   1. Initial program 80.0%

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

      1. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. *-commutative 48.6%

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

      5. times-frac 50.4%

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

      6. unpow2 50.4%

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

      7. *-commutative 50.4%

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

      8. unpow2 50.4%

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

      9. unpow2 50.4%

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

   6. Simplified 50.4%

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

      1. associate-*r/ 51.0%

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

      2. associate-/l* 55.5%

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

      3. *-commutative 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. add-cbrt-cube 55.5%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}}} \cdot -0.125\right) \]

      2. times-frac 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      3. associate-/r/ 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      4. associate-*r* 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      5. times-frac 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      6. associate-/r/ 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      7. associate-*r* 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      8. times-frac 62.2%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}} \cdot -0.125\right) \]

   10. Applied egg-rr 65.3%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)}} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. associate-*l* 65.3%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)}} \cdot -0.125\right) \]

       2. *-commutative 65.3%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)} \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       3. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       4. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       5. associate-*l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)}} \cdot -0.125\right) \]

       6. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       7. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       8. *-commutative 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)}\right)\right)} \cdot -0.125\right) \]

       9. associate-/l* 67.7%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)\right)\right)} \cdot -0.125\right) \]

       10. associate-/l* 71.4%

           \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right)\right)\right)} \cdot -0.125\right) \]

   12. Simplified 71.4%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right)}} \cdot -0.125\right) \]
   13. Step-by-step derivation

       1. *-commutative 71.4%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)}} \cdot -0.125\right) \]

       2. associate-*r* 71.5%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \frac{h \cdot M}{\frac{d}{M}}\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       3. *-commutative 71.5%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       4. add-cbrt-cube 71.5%

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

       5. frac-times 73.2%

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

   14. Applied egg-rr 73.2%

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

   if 1.01999999999999997e120 < d

   1. Initial program 86.9%

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

      1. times-frac 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in M around 0 84.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(h \cdot M\right) \cdot \frac{D}{\ell}}{\frac{d}{M} \cdot \frac{d}{D}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 76.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.7 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{D}{\ell} \cdot \left(M \cdot \frac{h \cdot M}{d}\right)}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.7e+119)
   (* w0 (+ 1.0 (* -0.125 (/ (* (/ D l) (* M (/ (* h M) d))) (/ d D)))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.7e+119) {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (d_1 <= 4.7d+119) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / l) * (m * ((h * m) / d_1))) / (d_1 / d))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.7e+119) {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.7e+119:
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.7e+119)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / l) * Float64(M * Float64(Float64(h * M) / d))) / Float64(d / D)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.7e+119)
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (M * ((h * M) / d))) / (d / D))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.7e+119], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 4.70000000000000008e119

   1. Initial program 80.0%

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

      1. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. *-commutative 48.6%

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

      5. times-frac 50.4%

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

      6. unpow2 50.4%

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

      7. *-commutative 50.4%

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

      8. unpow2 50.4%

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

      9. unpow2 50.4%

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

   6. Simplified 50.4%

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

      1. associate-*r/ 51.0%

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

      2. associate-/l* 55.5%

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

      3. *-commutative 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. add-cbrt-cube 55.5%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}}} \cdot -0.125\right) \]

      2. times-frac 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)} \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      3. associate-/r/ 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      4. associate-*r* 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      5. times-frac 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      6. associate-/r/ 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\color{blue}{\frac{D}{\ell} \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      7. associate-*r* 55.0%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{d}\right)\right) \cdot \frac{\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}} \cdot -0.125\right) \]

      8. times-frac 62.2%

         \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \color{blue}{\left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}} \cdot -0.125\right) \]

   10. Applied egg-rr 65.3%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)}} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. associate-*l* 65.3%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)}} \cdot -0.125\right) \]

       2. *-commutative 65.3%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)} \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       3. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       4. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right) \cdot \left(\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right) \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)} \cdot -0.125\right) \]

       5. associate-*l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \color{blue}{\left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)}} \cdot -0.125\right) \]

       6. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}} \cdot \left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       7. associate-/l* 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \left(\frac{\frac{D}{\ell} \cdot D}{d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}\right)\right)\right)} \cdot -0.125\right) \]

       8. *-commutative 64.8%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{\left(h \cdot M\right) \cdot M}{d} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)}\right)\right)} \cdot -0.125\right) \]

       9. associate-/l* 67.7%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\color{blue}{\frac{h \cdot M}{\frac{d}{M}}} \cdot \frac{\frac{D}{\ell} \cdot D}{d}\right)\right)\right)} \cdot -0.125\right) \]

       10. associate-/l* 71.4%

           \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \color{blue}{\frac{\frac{D}{\ell}}{\frac{d}{D}}}\right)\right)\right)} \cdot -0.125\right) \]

   12. Simplified 71.4%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right) \cdot \left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right)}} \cdot -0.125\right) \]
   13. Step-by-step derivation

       1. *-commutative 71.4%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)}} \cdot -0.125\right) \]

       2. associate-*r* 71.5%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\color{blue}{\left(\left(\frac{\frac{D}{\ell}}{\frac{d}{D}} \cdot \frac{h \cdot M}{\frac{d}{M}}\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       3. *-commutative 71.5%

          \[\leadsto w0 \cdot \left(1 + \sqrt[3]{\left(\color{blue}{\left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{h \cdot M}{\frac{d}{M}} \cdot \frac{\frac{D}{\ell}}{\frac{d}{D}}\right)} \cdot -0.125\right) \]

       4. add-cbrt-cube 71.5%

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

       5. associate-*r/ 73.9%

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

       6. associate-/r/ 73.9%

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

   14. Applied egg-rr 73.9%

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

   if 4.70000000000000008e119 < d

   1. Initial program 86.9%

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

      1. times-frac 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in M around 0 84.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.7 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{D}{\ell} \cdot \left(M \cdot \frac{h \cdot M}{d}\right)}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 72.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.02 \cdot 10^{+24}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 1.02e+24)
   w0
   (* -0.125 (* (* D (/ D l)) (/ w0 (* (/ d (* h M)) (/ d M)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.02e+24) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d / (h * M)) * (d / M))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 1.02d+24) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * (w0 / ((d_1 / (h * m)) * (d_1 / m))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.02e+24) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d / (h * M)) * (d / M))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.02e+24:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d / (h * M)) * (d / M))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.02e+24)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(w0 / Float64(Float64(d / Float64(h * M)) * Float64(d / M)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.02e+24)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d / (h * M)) * (d / M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.02e+24], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.02000000000000004e24

   1. Initial program 82.9%

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

      1. times-frac 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in M around 0 70.0%

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

   if 1.02000000000000004e24 < M

   1. Initial program 73.3%

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

      1. times-frac 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in M around 0 52.7%

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

      1. +-commutative 52.7%

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

      2. associate-*r/ 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-*r/ 52.7%

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

      5. *-commutative 52.7%

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

      6. fma-def 52.7%

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

      7. *-commutative 52.7%

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

      8. times-frac 52.7%

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

      9. unpow2 52.7%

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

      10. unpow2 52.7%

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

      11. times-frac 55.5%

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

      12. unpow2 55.5%

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

      13. *-commutative 55.5%

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

      14. unpow2 55.5%

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

      15. associate-*l* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in D around inf 29.8%

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

      1. associate-*r/ 29.8%

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

      2. unpow2 29.8%

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

      3. *-commutative 29.8%

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

      4. unpow2 29.8%

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

      5. *-commutative 29.8%

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

      6. associate-*r/ 29.8%

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

      7. times-frac 25.8%

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

      8. unpow2 25.8%

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

      9. unpow2 25.8%

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

      10. unpow2 25.8%

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

   9. Simplified 25.8%

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

   10. Taylor expanded in w0 around 0 25.8%

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

       1. *-commutative 25.8%

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

       2. associate-/l* 25.7%

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

       3. unpow2 25.7%

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

       4. *-commutative 25.7%

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

       5. unpow2 25.7%

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

       6. associate-*l* 25.7%

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

       7. times-frac 26.2%

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

   12. Simplified 26.2%

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

   13. Taylor expanded in D around 0 26.2%

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

       1. unpow2 26.2%

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

       2. associate-*l/ 26.2%

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

   15. Simplified 26.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.02 \cdot 10^{+24}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0}{\frac{d}{h \cdot M} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 12: 68.7% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 81.0%

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

   1. times-frac 80.7%

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

3. Simplified 80.7%

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

4. Taylor expanded in M around 0 65.9%

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

5. Final simplification 65.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023234 
(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))))))