Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 88.8%

Time: 14.3s

Alternatives: 13

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 D) (* 2.0 d)) 5e+84)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M D) (/ 0.5 d)) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* (/ D l) (* D (/ (* h -0.25) (* (/ d M) (/ d M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+84) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M * D) * (0.5 / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / 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: 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 * d) / (2.0d0 * d_1)) <= 5d+84) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * d) * (0.5d0 / d_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((d / l) * (d * ((h * (-0.25d0)) / ((d_1 / m) * (d_1 / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 * D) / (2.0 * d)) <= 5e+84) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M * D) * (0.5 / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 5e+84:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M * D) * (0.5 / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+84)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M * D) * Float64(0.5 / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D / l) * Float64(D * Float64(Float64(h * -0.25) / Float64(Float64(d / M) * Float64(d / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 5e+84)
		tmp = w0 * sqrt((1.0 - ((h * (((M * D) * (0.5 / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / 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: 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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+84], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D / l), $MachinePrecision] * N[(D * N[(N[(h * -0.25), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5.0000000000000001e84

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. times-frac 82.7%

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

   3. Simplified 82.7%

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

      1. *-commutative 82.7%

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

      2. frac-times 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*l/ 89.7%

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

      5. div-inv 89.7%

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

      6. associate-*l* 88.9%

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

      7. associate-/r* 88.9%

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

      8. metadata-eval 88.9%

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

   5. Applied egg-rr 88.9%

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

   6. Taylor expanded in M around 0 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r/ 89.7%

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

   8. Simplified 89.7%

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

   if 5.0000000000000001e84 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. times-frac 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in w0 around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. cancel-sign-sub-inv 55.9%

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

      3. *-commutative 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. *-commutative 55.9%

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

      6. cancel-sign-sub-inv 55.9%

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

      7. distribute-lft-neg-in 55.9%

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

      8. distribute-rgt-neg-in 55.9%

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

   6. Simplified 61.3%

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

      1. *-un-lft-identity 61.3%

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

      2. associate-*l* 61.3%

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

      3. associate-/r/ 61.3%

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

      4. times-frac 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. *-lft-identity 69.0%

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

      2. associate-*l* 69.0%

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

      3. associate-*l/ 69.0%

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

   10. Simplified 69.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \]

Alternative 2: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\left(\frac{d}{M} \cdot \frac{d}{M}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 0.01)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (/ (* D h) (* (* (/ d M) (/ d M)) (/ l D)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * h) / (((d / M) * (d / M)) * (l / D))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= 0.01d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * h) / (((d_1 / m) * (d_1 / m)) * (l / d))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		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 * ((D * h) / (((d / M) * (d / M)) * (l / D))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.01:
		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 * ((D * h) / (((d / M) * (d / M)) * (l / D))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.01)
		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(-0.25 * Float64(Float64(D * h) / Float64(Float64(Float64(d / M) * Float64(d / M)) * Float64(l / D)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= 0.01)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * h) / (((d / M) * (d / M)) * (l / D))))));
	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: 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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.01], 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[(-0.25 * N[(N[(D * h), $MachinePrecision] / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < 0.0100000000000000002

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

      2. times-frac 88.1%

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

   3. Simplified 88.1%

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

   if 0.0100000000000000002 < (*.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. *-commutative 0.0%

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

      2. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in w0 around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. cancel-sign-sub-inv 48.3%

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

      3. *-commutative 48.3%

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

      4. cancel-sign-sub-inv 48.3%

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

      5. *-commutative 48.3%

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

      6. cancel-sign-sub-inv 48.3%

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

      7. distribute-lft-neg-in 48.3%

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

      8. distribute-rgt-neg-in 48.3%

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

   6. Simplified 57.5%

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

      1. frac-times 66.2%

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

      2. times-frac 78.6%

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\left(\frac{d}{M} \cdot \frac{d}{M}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \]

Alternative 3: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (/ (* M D) (* 2.0 d)) 2.0) 2e+66)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* (/ D l) (* D (/ (* h -0.25) (* (/ d M) (/ d M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 2e+66) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / 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: 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 * d) / (2.0d0 * d_1)) ** 2.0d0) <= 2d+66) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((d / l) * (d * ((h * (-0.25d0)) / ((d_1 / m) * (d_1 / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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(((M * D) / (2.0 * d)), 2.0) <= 2e+66) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+66:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 2e+66)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D / l) * Float64(D * Float64(Float64(h * -0.25) / Float64(Float64(d / M) * Float64(d / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 2e+66)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / 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: 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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+66], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D / l), $MachinePrecision] * N[(D * N[(N[(h * -0.25), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 1.99999999999999989e66

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. frac-times 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-*l/ 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*l* 98.8%

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

      7. associate-/r* 98.8%

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

      8. metadata-eval 98.8%

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

   5. Applied egg-rr 98.8%

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

   if 1.99999999999999989e66 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

      2. times-frac 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in w0 around 0 41.2%

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

      1. *-commutative 41.2%

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

      2. cancel-sign-sub-inv 41.2%

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

      3. *-commutative 41.2%

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

      4. cancel-sign-sub-inv 41.2%

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

      5. *-commutative 41.2%

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

      6. cancel-sign-sub-inv 41.2%

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

      7. distribute-lft-neg-in 41.2%

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

      8. distribute-rgt-neg-in 41.2%

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

   6. Simplified 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. associate-*l* 48.4%

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

      3. associate-/r/ 48.4%

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

      4. times-frac 58.7%

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

   8. Applied egg-rr 58.7%

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

      1. *-lft-identity 58.7%

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

      2. associate-*l* 63.4%

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

      3. associate-*l/ 63.4%

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

   10. Simplified 63.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+66}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \end{array} \]

Alternative 4: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-110} \lor \neg \left(\ell \leq 10^{-109}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{\frac{M \cdot h}{\frac{\ell}{M}}}{d}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (or (<= l -5e-110) (not (<= l 1e-109)))
   (* w0 (sqrt (+ 1.0 (* (/ D l) (* D (/ (* h -0.25) (* (/ d M) (/ d M))))))))
   (* w0 (+ 1.0 (* (* (* D (/ D d)) (/ (/ (* M h) (/ l M)) d)) -0.125)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((l <= -5e-110) || !(l <= 1e-109)) {
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / 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: 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 ((l <= (-5d-110)) .or. (.not. (l <= 1d-109))) then
        tmp = w0 * sqrt((1.0d0 + ((d / l) * (d * ((h * (-0.25d0)) / ((d_1 / m) * (d_1 / m)))))))
    else
        tmp = w0 * (1.0d0 + (((d * (d / d_1)) * (((m * h) / (l / m)) / d_1)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 ((l <= -5e-110) || !(l <= 1e-109)) {
		tmp = w0 * Math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / d)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (l <= -5e-110) or not (l <= 1e-109):
		tmp = w0 * math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))))
	else:
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / d)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((l <= -5e-110) || !(l <= 1e-109))
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D / l) * Float64(D * Float64(Float64(h * -0.25) / Float64(Float64(d / M) * Float64(d / M))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D / d)) * Float64(Float64(Float64(M * h) / Float64(l / M)) / d)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((l <= -5e-110) || ~((l <= 1e-109)))
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	else
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / 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: 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[Or[LessEqual[l, -5e-110], N[Not[LessEqual[l, 1e-109]], $MachinePrecision]], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D / l), $MachinePrecision] * N[(D * N[(N[(h * -0.25), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5e-110 or 9.9999999999999999e-110 < l

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. times-frac 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in w0 around 0 51.9%

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

      1. *-commutative 51.9%

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

      2. cancel-sign-sub-inv 51.9%

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

      3. *-commutative 51.9%

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

      4. cancel-sign-sub-inv 51.9%

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

      5. *-commutative 51.9%

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

      6. cancel-sign-sub-inv 51.9%

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

      7. distribute-lft-neg-in 51.9%

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

      8. distribute-rgt-neg-in 51.9%

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

   6. Simplified 59.2%

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

      1. *-un-lft-identity 59.2%

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

      2. associate-*l* 59.2%

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

      3. associate-/r/ 59.2%

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

      4. times-frac 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. *-lft-identity 73.4%

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

      2. associate-*l* 80.5%

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

      3. associate-*l/ 80.5%

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

   10. Simplified 80.5%

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

   if -5e-110 < l < 9.9999999999999999e-110

   1. Initial program 66.6%

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

      1. *-commutative 66.6%

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

      2. times-frac 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in M around 0 57.5%

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

      1. *-commutative 57.5%

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

      2. *-commutative 57.5%

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

      3. times-frac 57.6%

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

      4. *-commutative 57.6%

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

      5. unpow2 57.6%

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

      6. unpow2 57.6%

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

      7. *-commutative 57.6%

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

      8. unpow2 57.6%

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

   6. Simplified 57.6%

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

   7. Taylor expanded in D around 0 57.5%

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

      1. *-commutative 57.5%

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

      2. times-frac 57.6%

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

      3. unpow2 57.6%

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

      4. *-commutative 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-/l* 46.7%

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

      7. associate-*l/ 46.7%

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

      8. unpow2 46.7%

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

      9. times-frac 51.9%

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

      10. unpow2 51.9%

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

      11. associate-*r/ 54.3%

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

      12. associate-/r/ 60.5%

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

      13. *-commutative 60.5%

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

      14. associate-/l* 61.9%

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

   9. Simplified 61.9%

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

       1. associate-*r/ 64.2%

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

   11. Applied egg-rr 64.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-110} \lor \neg \left(\ell \leq 10^{-109}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{\frac{M \cdot h}{\frac{\ell}{M}}}{d}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 1.45e+18)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ D (/ l D)) (* M (/ (* M (/ h d)) d)))))))
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.45e+18) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * (M * ((M * (h / d)) / d))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 1.45d+18) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (l / d)) * (m * ((m * (h / d_1)) / d_1))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((h * (m * m)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 1.45e+18) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * (M * ((M * (h / d)) / d))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.45e+18:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * (M * ((M * (h / d)) / d))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.45e+18)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(l / D)) * Float64(M * Float64(Float64(M * Float64(h / d)) / d)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.45e+18)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * (M * ((M * (h / d)) / d))))));
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / 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: 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, 1.45e+18], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.45e18

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. times-frac 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in w0 around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. cancel-sign-sub-inv 50.5%

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

      3. *-commutative 50.5%

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

      4. cancel-sign-sub-inv 50.5%

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

      5. *-commutative 50.5%

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

      6. cancel-sign-sub-inv 50.5%

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

      7. distribute-lft-neg-in 50.5%

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

      8. distribute-rgt-neg-in 50.5%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in h around 0 54.2%

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

      1. unpow2 54.2%

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

      2. associate-*l/ 54.4%

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

      3. unpow2 54.4%

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

      4. associate-*r* 59.3%

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

      5. *-commutative 59.3%

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

      6. associate-/r* 66.9%

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

      7. associate-*l/ 70.3%

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

   9. Simplified 70.3%

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

   if 1.45e18 < d

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

      2. times-frac 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in M around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. *-commutative 63.9%

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

      3. times-frac 60.4%

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

      4. *-commutative 60.4%

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

      5. unpow2 60.4%

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

      6. unpow2 60.4%

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

      7. *-commutative 60.4%

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

      8. unpow2 60.4%

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

   6. Simplified 60.4%

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

   7. Taylor expanded in D around 0 60.4%

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

      1. unpow2 2.7%

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

      2. unpow2 2.7%

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

      3. times-frac 5.2%

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

   9. Simplified 73.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \]

Alternative 6: 85.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 4.2e+18)
   (* w0 (sqrt (+ 1.0 (* (/ D l) (* D (/ (* h -0.25) (* (/ d M) (/ d M))))))))
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (* (/ D d) (/ D d)) (* h (* M M)))) l))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.2e+18) {
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * (((D / d) * (D / d)) * (h * (M * M)))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 4.2d+18) then
        tmp = w0 * sqrt((1.0d0 + ((d / l) * (d * ((h * (-0.25d0)) / ((d_1 / m) * (d_1 / m)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (((d / d_1) * (d / d_1)) * (h * (m * m)))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 4.2e+18) {
		tmp = w0 * Math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (((D / d) * (D / d)) * (h * (M * M)))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.2e+18:
		tmp = w0 * math.sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (((D / d) * (D / d)) * (h * (M * M)))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.2e+18)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D / l) * Float64(D * Float64(Float64(h * -0.25) / Float64(Float64(d / M) * Float64(d / M))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M * M)))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.2e+18)
		tmp = w0 * sqrt((1.0 + ((D / l) * (D * ((h * -0.25) / ((d / M) * (d / M)))))));
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * (((D / d) * (D / d)) * (h * (M * M)))) / 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: 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, 4.2e+18], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D / l), $MachinePrecision] * N[(D * N[(N[(h * -0.25), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.2e18

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. times-frac 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in w0 around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. cancel-sign-sub-inv 50.5%

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

      3. *-commutative 50.5%

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

      4. cancel-sign-sub-inv 50.5%

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

      5. *-commutative 50.5%

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

      6. cancel-sign-sub-inv 50.5%

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

      7. distribute-lft-neg-in 50.5%

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

      8. distribute-rgt-neg-in 50.5%

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

   6. Simplified 55.9%

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

      1. *-un-lft-identity 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-/r/ 55.9%

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

      4. times-frac 70.3%

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

   8. Applied egg-rr 70.3%

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

      1. *-lft-identity 70.3%

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

      2. associate-*l* 75.9%

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

      3. associate-*l/ 75.9%

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

   10. Simplified 75.9%

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

   if 4.2e18 < d

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

      2. times-frac 91.0%

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

   3. Simplified 91.0%

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

      1. *-commutative 91.0%

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

      2. frac-times 89.2%

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

      3. *-commutative 89.2%

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

      4. associate-*l/ 94.7%

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

      5. div-inv 94.7%

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

      6. associate-*l* 94.7%

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

      7. associate-/r* 94.7%

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

      8. metadata-eval 94.7%

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

   5. Applied egg-rr 94.7%

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

   6. Taylor expanded in h around 0 65.6%

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

      1. associate-/l* 67.4%

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

      2. unpow2 67.4%

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

      3. associate-/r/ 67.3%

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

      4. unpow2 67.3%

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

      5. unpow2 67.3%

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

      6. times-frac 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{\ell} \cdot \left(D \cdot \frac{h \cdot -0.25}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}}\\ \end{array} \]

Alternative 7: 71.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{w0}{\ell} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 7e-39)
   w0
   (if (<= M 3.4e-16)
     (* -0.125 (* (/ (* D D) (* d d)) (* (/ w0 l) (* M (* M h)))))
     (if (<= M 2.05e+20)
       w0
       (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 (* h (* M M))) l)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7e-39) {
		tmp = w0;
	} else if (M <= 3.4e-16) {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 / l) * (M * (M * h))));
	} else if (M <= 2.05e+20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 7d-39) then
        tmp = w0
    else if (m <= 3.4d-16) then
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * ((w0 / l) * (m * (m * h))))
    else if (m <= 2.05d+20) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * (h * (m * m))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 7e-39) {
		tmp = w0;
	} else if (M <= 3.4e-16) {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 / l) * (M * (M * h))));
	} else if (M <= 2.05e+20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7e-39:
		tmp = w0
	elif M <= 3.4e-16:
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 / l) * (M * (M * h))))
	elif M <= 2.05e+20:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7e-39)
		tmp = w0;
	elseif (M <= 3.4e-16)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(w0 / l) * Float64(M * Float64(M * h)))));
	elseif (M <= 2.05e+20)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7e-39)
		tmp = w0;
	elseif (M <= 3.4e-16)
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 / l) * (M * (M * h))));
	elseif (M <= 2.05e+20)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / 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: 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, 7e-39], w0, If[LessEqual[M, 3.4e-16], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 / l), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.05e+20], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.99999999999999999e-39 or 3.4e-16 < M < 2.05e20

   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. *-commutative 80.0%

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

      2. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in M around 0 71.9%

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

   if 6.99999999999999999e-39 < M < 3.4e-16

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. times-frac 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in M around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. *-commutative 61.4%

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

      3. times-frac 61.4%

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

      4. *-commutative 61.4%

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

      5. unpow2 61.4%

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

      6. unpow2 61.4%

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

      7. *-commutative 61.4%

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

      8. unpow2 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in D around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. times-frac 61.4%

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

      3. unpow2 61.4%

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

      4. *-commutative 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-/l* 61.4%

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

      7. associate-*l/ 61.4%

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

      8. unpow2 61.4%

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

      9. times-frac 62.0%

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

      10. unpow2 62.0%

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

      11. associate-*r/ 62.0%

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

      12. associate-/r/ 62.0%

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

      13. *-commutative 62.0%

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

      14. associate-/l* 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in D around inf 21.9%

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

       1. associate-*r/ 21.9%

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

       2. *-commutative 21.9%

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

       3. *-commutative 21.9%

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

       4. associate-*r/ 21.9%

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

       5. times-frac 21.9%

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

       6. unpow2 21.9%

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

       7. unpow2 21.9%

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

       8. *-commutative 21.9%

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

       9. unpow2 21.9%

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

   12. Simplified 21.9%

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

   13. Taylor expanded in h around 0 21.9%

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

       1. associate-/l* 21.9%

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

       2. *-commutative 21.9%

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

       3. associate-/r/ 21.9%

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

       4. unpow2 21.9%

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

       5. associate-*r* 21.9%

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

       6. *-commutative 21.9%

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

   15. Simplified 21.9%

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

   if 2.05e20 < M

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. times-frac 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in M around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 56.8%

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

      4. *-commutative 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. *-commutative 56.8%

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

      8. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in D around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. times-frac 56.8%

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

      3. unpow2 56.8%

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

      4. *-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-/l* 58.9%

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

      7. associate-*l/ 58.8%

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

      8. unpow2 58.8%

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

      9. times-frac 61.3%

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

      10. unpow2 61.3%

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

      11. associate-*r/ 65.7%

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

      12. associate-/r/ 57.1%

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

      13. *-commutative 57.1%

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

      14. associate-/l* 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in D around inf 33.8%

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

       1. associate-*r/ 33.8%

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

       2. *-commutative 33.8%

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

       3. *-commutative 33.8%

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

       4. associate-*r/ 33.8%

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

       5. times-frac 33.8%

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

       6. unpow2 33.8%

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

       7. unpow2 33.8%

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

       8. *-commutative 33.8%

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

       9. unpow2 33.8%

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

   12. Simplified 33.8%

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

   13. Taylor expanded in D around 0 33.8%

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

       1. unpow2 33.8%

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

       2. unpow2 33.8%

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

       3. times-frac 36.4%

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

   15. Simplified 36.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{w0}{\ell} \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 8: 71.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;M \leq 7 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \frac{t_0}{\frac{\ell}{w0}}}{d \cdot d}\\ \mathbf{elif}\;M \leq 9 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot t_0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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
 (let* ((t_0 (* h (* M M))))
   (if (<= M 7e-39)
     w0
     (if (<= M 3.4e-16)
       (* -0.125 (/ (* (* D D) (/ t_0 (/ l w0))) (* d d)))
       (if (<= M 9e+18)
         w0
         (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 t_0) l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = h * (M * M);
	double tmp;
	if (M <= 7e-39) {
		tmp = w0;
	} else if (M <= 3.4e-16) {
		tmp = -0.125 * (((D * D) * (t_0 / (l / w0))) / (d * d));
	} else if (M <= 9e+18) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * t_0) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) :: t_0
    real(8) :: tmp
    t_0 = h * (m * m)
    if (m <= 7d-39) then
        tmp = w0
    else if (m <= 3.4d-16) then
        tmp = (-0.125d0) * (((d * d) * (t_0 / (l / w0))) / (d_1 * d_1))
    else if (m <= 9d+18) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * t_0) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = h * (M * M);
	double tmp;
	if (M <= 7e-39) {
		tmp = w0;
	} else if (M <= 3.4e-16) {
		tmp = -0.125 * (((D * D) * (t_0 / (l / w0))) / (d * d));
	} else if (M <= 9e+18) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * t_0) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = h * (M * M)
	tmp = 0
	if M <= 7e-39:
		tmp = w0
	elif M <= 3.4e-16:
		tmp = -0.125 * (((D * D) * (t_0 / (l / w0))) / (d * d))
	elif M <= 9e+18:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * t_0) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (M <= 7e-39)
		tmp = w0;
	elseif (M <= 3.4e-16)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) * Float64(t_0 / Float64(l / w0))) / Float64(d * d)));
	elseif (M <= 9e+18)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * t_0) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (M <= 7e-39)
		tmp = w0;
	elseif (M <= 3.4e-16)
		tmp = -0.125 * (((D * D) * (t_0 / (l / w0))) / (d * d));
	elseif (M <= 9e+18)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * t_0) / 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: 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_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 7e-39], w0, If[LessEqual[M, 3.4e-16], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[(t$95$0 / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 9e+18], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.99999999999999999e-39 or 3.4e-16 < M < 9e18

   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. *-commutative 80.0%

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

      2. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in M around 0 71.9%

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

   if 6.99999999999999999e-39 < M < 3.4e-16

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. times-frac 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in M around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. *-commutative 61.4%

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

      3. times-frac 61.4%

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

      4. *-commutative 61.4%

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

      5. unpow2 61.4%

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

      6. unpow2 61.4%

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

      7. *-commutative 61.4%

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

      8. unpow2 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in D around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. times-frac 61.4%

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

      3. unpow2 61.4%

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

      4. *-commutative 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-/l* 61.4%

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

      7. associate-*l/ 61.4%

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

      8. unpow2 61.4%

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

      9. times-frac 62.0%

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

      10. unpow2 62.0%

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

      11. associate-*r/ 62.0%

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

      12. associate-/r/ 62.0%

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

      13. *-commutative 62.0%

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

      14. associate-/l* 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in D around inf 21.9%

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

       1. associate-*r/ 21.9%

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

       2. *-commutative 21.9%

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

       3. *-commutative 21.9%

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

       4. associate-*r/ 21.9%

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

       5. times-frac 21.9%

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

       6. unpow2 21.9%

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

       7. unpow2 21.9%

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

       8. *-commutative 21.9%

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

       9. unpow2 21.9%

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

   12. Simplified 21.9%

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

       1. associate-*l/ 21.9%

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

       2. associate-/l* 21.9%

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

   14. Applied egg-rr 21.9%

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

   if 9e18 < M

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. times-frac 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in M around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 56.8%

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

      4. *-commutative 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. *-commutative 56.8%

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

      8. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in D around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. times-frac 56.8%

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

      3. unpow2 56.8%

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

      4. *-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-/l* 58.9%

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

      7. associate-*l/ 58.8%

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

      8. unpow2 58.8%

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

      9. times-frac 61.3%

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

      10. unpow2 61.3%

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

      11. associate-*r/ 65.7%

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

      12. associate-/r/ 57.1%

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

      13. *-commutative 57.1%

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

      14. associate-/l* 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in D around inf 33.8%

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

       1. associate-*r/ 33.8%

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

       2. *-commutative 33.8%

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

       3. *-commutative 33.8%

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

       4. associate-*r/ 33.8%

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

       5. times-frac 33.8%

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

       6. unpow2 33.8%

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

       7. unpow2 33.8%

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

       8. *-commutative 33.8%

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

       9. unpow2 33.8%

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

   12. Simplified 33.8%

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

   13. Taylor expanded in D around 0 33.8%

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

       1. unpow2 33.8%

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

       2. unpow2 33.8%

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

       3. times-frac 36.4%

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

   15. Simplified 36.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\frac{\ell}{w0}}}{d \cdot d}\\ \mathbf{elif}\;M \leq 9 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 9: 76.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-234}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (<= (/ h l) -1e-234)
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D d)) (/ (* h (/ M (/ l M))) d)))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-234) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h * (M / (l / M))) / 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: 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 ((h / l) <= (-1d-234)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / d_1)) * ((h * (m / (l / m))) / d_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 ((h / l) <= -1e-234) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h * (M / (l / M))) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-234:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h * (M / (l / M))) / d))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-234)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / d)) * Float64(Float64(h * Float64(M / Float64(l / M))) / d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-234)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h * (M / (l / M))) / 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: 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[(h / l), $MachinePrecision], -1e-234], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.9999999999999996e-235

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. times-frac 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in M around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 50.7%

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

      4. *-commutative 50.7%

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

      5. unpow2 50.7%

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

      6. unpow2 50.7%

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

      7. *-commutative 50.7%

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

      8. unpow2 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in D around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. times-frac 50.7%

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

      3. unpow2 50.7%

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

      4. *-commutative 50.7%

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

      5. *-commutative 50.7%

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

      6. associate-/l* 47.1%

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

      7. associate-*l/ 47.1%

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

      8. unpow2 47.1%

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

      9. times-frac 55.5%

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

      10. unpow2 55.5%

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

      11. associate-*r/ 59.2%

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

      12. associate-/r/ 62.1%

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

      13. *-commutative 62.1%

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

      14. associate-/l* 63.7%

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

   9. Simplified 63.7%

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

   if -9.9999999999999996e-235 < (/.f64 h l)

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. times-frac 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in M around 0 90.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-234}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 77.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-234}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{\frac{M \cdot h}{\frac{\ell}{M}}}{d}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (<= (/ h l) -1e-234)
   (* w0 (+ 1.0 (* (* (* D (/ D d)) (/ (/ (* M h) (/ l M)) d)) -0.125)))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-234) {
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / 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: 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 ((h / l) <= (-1d-234)) then
        tmp = w0 * (1.0d0 + (((d * (d / d_1)) * (((m * h) / (l / m)) / d_1)) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 ((h / l) <= -1e-234) {
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / d)) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-234:
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / d)) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-234)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D / d)) * Float64(Float64(Float64(M * h) / Float64(l / M)) / d)) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-234)
		tmp = w0 * (1.0 + (((D * (D / d)) * (((M * h) / (l / M)) / 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: 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[(h / l), $MachinePrecision], -1e-234], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.9999999999999996e-235

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. times-frac 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in M around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 50.7%

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

      4. *-commutative 50.7%

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

      5. unpow2 50.7%

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

      6. unpow2 50.7%

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

      7. *-commutative 50.7%

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

      8. unpow2 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in D around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. times-frac 50.7%

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

      3. unpow2 50.7%

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

      4. *-commutative 50.7%

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

      5. *-commutative 50.7%

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

      6. associate-/l* 47.1%

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

      7. associate-*l/ 47.1%

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

      8. unpow2 47.1%

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

      9. times-frac 55.5%

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

      10. unpow2 55.5%

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

      11. associate-*r/ 59.2%

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

      12. associate-/r/ 62.1%

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

      13. *-commutative 62.1%

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

      14. associate-/l* 63.7%

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

   9. Simplified 63.7%

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

       1. associate-*r/ 62.2%

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

   11. Applied egg-rr 62.2%

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

   if -9.9999999999999996e-235 < (/.f64 h l)

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. times-frac 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in M around 0 90.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-234}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{d}\right) \cdot \frac{\frac{M \cdot h}{\frac{\ell}{M}}}{d}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 1.15e-45)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D d)) (* (/ h l) (/ M (/ d M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.15e-45) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h / l) * (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: 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 <= 1.15d-45) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / d_1)) * ((h / l) * (m / (d_1 / m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 1.15e-45) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h / l) * (M / (d / M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.15e-45:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h / l) * (M / (d / M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.15e-45)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / d)) * Float64(Float64(h / l) * Float64(M / Float64(d / M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.15e-45)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / d)) * ((h / l) * (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: 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, 1.15e-45], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.14999999999999996e-45

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in M around 0 71.3%

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

   if 1.14999999999999996e-45 < M

   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. *-commutative 81.3%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in M around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. *-commutative 57.2%

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

      3. times-frac 53.9%

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

      4. *-commutative 53.9%

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

      5. unpow2 53.9%

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

      6. unpow2 53.9%

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

      7. *-commutative 53.9%

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

      8. unpow2 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in D around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. times-frac 53.9%

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

      3. unpow2 53.9%

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

      4. *-commutative 53.9%

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

      5. *-commutative 53.9%

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

      6. associate-/l* 55.5%

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

      7. associate-*l/ 55.4%

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

      8. unpow2 55.4%

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

      9. times-frac 63.8%

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

      10. unpow2 63.8%

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

      11. associate-*r/ 67.0%

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

      12. associate-/r/ 60.7%

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

      13. *-commutative 60.7%

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

      14. associate-/l* 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in h around 0 67.1%

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

       1. *-commutative 67.1%

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

       2. times-frac 67.0%

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

       3. unpow2 67.0%

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

       4. associate-/l* 71.8%

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

   12. Simplified 71.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 12: 72.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 2.25e+20)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 (* h (* M M))) l)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.25e+20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 2.25d+20) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * (h * (m * m))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.25e+20) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.25e+20:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.25e+20)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.25e+20)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / 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: 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, 2.25e+20], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.25e20

   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. *-commutative 79.6%

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

      2. times-frac 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in M around 0 71.2%

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

   if 2.25e20 < M

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. times-frac 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in M around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 56.8%

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

      4. *-commutative 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. *-commutative 56.8%

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

      8. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in D around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. times-frac 56.8%

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

      3. unpow2 56.8%

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

      4. *-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-/l* 58.9%

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

      7. associate-*l/ 58.8%

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

      8. unpow2 58.8%

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

      9. times-frac 61.3%

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

      10. unpow2 61.3%

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

      11. associate-*r/ 65.7%

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

      12. associate-/r/ 57.1%

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

      13. *-commutative 57.1%

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

      14. associate-/l* 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in D around inf 33.8%

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

       1. associate-*r/ 33.8%

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

       2. *-commutative 33.8%

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

       3. *-commutative 33.8%

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

       4. associate-*r/ 33.8%

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

       5. times-frac 33.8%

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

       6. unpow2 33.8%

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

       7. unpow2 33.8%

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

       8. *-commutative 33.8%

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

       9. unpow2 33.8%

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

   12. Simplified 33.8%

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

   13. Taylor expanded in D around 0 33.8%

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

       1. unpow2 33.8%

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

       2. unpow2 33.8%

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

       3. times-frac 36.4%

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

   15. Simplified 36.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 13: 68.6% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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);
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: 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);
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)
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)
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)
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: 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 80.6%

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

   1. *-commutative 80.6%

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

   2. times-frac 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in M around 0 66.7%

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

5. Final simplification 66.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023207 
(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))))))