Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 88.1%

Time: 13.3s

Alternatives: 12

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+61}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e+61)
   (* w0 (sqrt (+ 1.0 (* (* h (pow (* M (/ (/ D 2.0) d)) 2.0)) (/ -1.0 l)))))
   (*
    w0
    (sqrt (+ 1.0 (* (/ (* D h) (* (/ l D) (* (/ d M) (/ d M)))) -0.25))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61) {
		tmp = w0 * sqrt((1.0 + ((h * pow((M * ((D / 2.0) / d)), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 1d+61) then
        tmp = w0 * sqrt((1.0d0 + ((h * ((m * ((d / 2.0d0) / d_1)) ** 2.0d0)) * ((-1.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + (((d * h) / ((l / d) * ((d_1 / m) * (d_1 / m)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61) {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow((M * ((D / 2.0) / d)), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow((M * ((D / 2.0) / d)), 2.0)) * (-1.0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 1e+61)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(M * Float64(Float64(D / 2.0) / d)) ^ 2.0)) * Float64(-1.0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M)))) * -0.25))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 1e+61)
		tmp = w0 * sqrt((1.0 + ((h * ((M * ((D / 2.0) / d)) ^ 2.0)) * (-1.0 / l))));
	else
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+61], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(M * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.99999999999999949e60

   1. Initial program 92.5%

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

      1. *-commutative 92.5%

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

      2. times-frac 92.5%

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

   3. Simplified 92.5%

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

      1. associate-*r/ 99.4%

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

      2. frac-times 99.4%

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

      3. *-commutative 99.4%

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

      4. clear-num 99.4%

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

      5. *-commutative 99.4%

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

      6. div-inv 99.4%

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

      7. associate-*l* 99.9%

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

      8. associate-/r* 99.9%

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

      9. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-/r/ 99.9%

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

      7. associate-*r/ 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. associate-/l* 99.9%

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

      10. associate-*r/ 99.9%

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

      11. associate-/r* 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   if 9.99999999999999949e60 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. times-frac 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in w0 around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. cancel-sign-sub-inv 36.2%

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

      3. *-commutative 36.2%

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

      4. cancel-sign-sub-inv 36.2%

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

      5. *-commutative 36.2%

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

      6. cancel-sign-sub-inv 36.2%

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

      7. distribute-lft-neg-in 36.2%

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

      8. distribute-rgt-neg-in 36.2%

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

   6. Simplified 45.8%

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

      1. frac-times 51.8%

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

      2. times-frac 66.3%

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

   8. Applied egg-rr 66.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+61}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.25}\\ \end{array} \]

Alternative 2: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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)) 4e-15)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (* w0 (+ 1.0 (* (* D (/ (/ D (/ (* (/ d M) (/ d M)) h)) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 4e-15) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= 4d-15) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / d_1)) ** 2.0d0))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d / (((d_1 / m) * (d_1 / m)) / h)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 4e-15) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 4e-15:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 4e-15)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= 4e-15)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 4e-15], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $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)) < 4.0000000000000003e-15

   1. Initial program 89.9%

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

      1. *-commutative 89.9%

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

      2. times-frac 89.0%

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

   3. Simplified 89.0%

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

   if 4.0000000000000003e-15 < (*.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 14.4%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in M around 0 38.5%

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

      1. *-commutative 38.5%

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

      2. *-commutative 38.5%

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

      3. times-frac 38.5%

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

      4. *-commutative 38.5%

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

      5. unpow2 38.5%

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

      6. unpow2 38.5%

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

      7. *-commutative 38.5%

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

      8. unpow2 38.5%

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

   6. Simplified 38.5%

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

   7. Taylor expanded in D around 0 38.5%

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

      1. times-frac 38.5%

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

      2. unpow2 38.5%

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

      3. associate-*r/ 48.0%

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

      4. unpow2 48.0%

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

      5. *-commutative 48.0%

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

      6. associate-*l/ 52.7%

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

      7. unpow2 52.7%

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

      8. associate-*l* 57.5%

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

      9. *-commutative 57.5%

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

      10. associate-*r/ 67.1%

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

      11. *-commutative 67.1%

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

      12. associate-*l* 67.1%

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

      13. *-commutative 67.1%

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

      14. unpow2 67.1%

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

      15. associate-*r/ 57.5%

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

      16. unpow2 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in M around 0 62.3%

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

       1. associate-/l* 62.3%

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

       2. unpow2 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-/r* 67.1%

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

       5. unpow2 67.1%

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

       6. times-frac 72.6%

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

       7. unpow2 72.6%

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

   12. Simplified 72.6%

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

       1. unpow2 72.6%

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

   14. Applied egg-rr 72.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 3: 86.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M * N[(N[(D / d), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d / M), $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) < +inf.0

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 87.6%

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

      2. frac-times 87.6%

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

      3. *-commutative 87.6%

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

      4. clear-num 87.6%

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

      5. *-commutative 87.6%

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

      6. div-inv 87.6%

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

      7. associate-*l* 89.5%

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

      8. associate-/r* 89.5%

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

      9. metadata-eval 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. associate-/r/ 89.5%

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

      2. associate-*r/ 89.5%

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

      3. *-commutative 89.5%

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

      4. associate-*r/ 89.5%

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

      5. metadata-eval 89.5%

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

      6. associate-/r/ 89.5%

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

      7. associate-*r/ 89.5%

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

      8. *-rgt-identity 89.5%

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

      9. associate-/l* 89.5%

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

      10. associate-*r/ 89.5%

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

      11. associate-/r* 89.5%

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

      12. *-commutative 89.5%

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

      13. associate-/r* 89.5%

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

   7. Simplified 89.5%

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

      1. *-un-lft-identity 89.5%

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

      2. associate-*l/ 89.5%

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

      3. *-un-lft-identity 89.5%

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

      4. associate-/l/ 89.5%

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

   9. Applied egg-rr 89.5%

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

       1. *-lft-identity 89.5%

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

       2. associate-/l* 89.8%

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

       3. associate-*r/ 88.3%

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

       4. *-commutative 88.3%

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

       5. *-commutative 88.3%

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

   11. Simplified 88.3%

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

   if +inf.0 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   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 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w0 around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. cancel-sign-sub-inv 0.0%

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

      3. *-commutative 0.0%

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

      4. cancel-sign-sub-inv 0.0%

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

      5. *-commutative 0.0%

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

      6. cancel-sign-sub-inv 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

   6. Simplified 0.0%

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

      1. associate-*r/ 0.0%

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

      2. associate-/r/ 0.0%

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

      3. times-frac 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. times-frac 100.0%

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

      2. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in D around 0 0.0%

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

       1. associate-/l/ 2.0%

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

       2. associate-*l/ 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r/ 100.0%

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

       5. *-commutative 100.0%

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

       6. associate-*r/ 8.4%

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

       7. associate-/l* 8.4%

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

       8. associate-/r/ 8.4%

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

   13. Simplified 8.4%

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

4. Final simplification 88.0%

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

Alternative 4: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+61}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e+61)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* (/ (* D h) (* (/ l D) (* (/ d M) (/ d M)))) -0.25))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 1d+61) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + (((d * h) / ((l / d) * ((d_1 / m) * (d_1 / m)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61) {
		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 * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+61:
		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 * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 1e+61)
		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(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M)))) * -0.25))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 1e+61)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * ((d / M) * (d / M)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+61], 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[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.99999999999999949e60

   1. Initial program 92.5%

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

      1. *-commutative 92.5%

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

      2. times-frac 92.5%

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

   3. Simplified 92.5%

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

      1. *-commutative 92.5%

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

      2. frac-times 92.5%

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

      3. *-commutative 92.5%

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

      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* 99.9%

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

      7. associate-/r* 99.9%

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

      8. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 9.99999999999999949e60 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. times-frac 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in w0 around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. cancel-sign-sub-inv 36.2%

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

      3. *-commutative 36.2%

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

      4. cancel-sign-sub-inv 36.2%

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

      5. *-commutative 36.2%

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

      6. cancel-sign-sub-inv 36.2%

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

      7. distribute-lft-neg-in 36.2%

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

      8. distribute-rgt-neg-in 36.2%

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

   6. Simplified 45.8%

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

      1. frac-times 51.8%

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

      2. times-frac 66.3%

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

   8. Applied egg-rr 66.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+61}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.25}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+103}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{D}{\ell}\right)\right) \cdot \frac{h}{\frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= h -2e+103)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* M (* (/ D d) (/ D l))) (/ h (/ d M)))))))
   (* w0 (+ 1.0 (* (* D (/ (/ D (/ (* (/ d M) (/ d M)) h)) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -2e+103) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M * ((D / d) * (D / l))) * (h / (d / M))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-2d+103)) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m * ((d / d_1) * (d / l))) * (h / (d_1 / m))))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d / (((d_1 / m) * (d_1 / m)) / h)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (h <= -2e+103) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M * ((D / d) * (D / l))) * (h / (d / M))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if h <= -2e+103:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M * ((D / d) * (D / l))) * (h / (d / M))))))
	else:
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (h <= -2e+103)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M * Float64(Float64(D / d) * Float64(D / l))) * Float64(h / Float64(d / M)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (h <= -2e+103)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M * ((D / d) * (D / l))) * (h / (d / M))))));
	else
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[h, -2e+103], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M * N[(N[(D / d), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2e103

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. times-frac 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in w0 around 0 58.1%

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

      1. *-commutative 58.1%

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

      2. cancel-sign-sub-inv 58.1%

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

      3. *-commutative 58.1%

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

      4. cancel-sign-sub-inv 58.1%

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

      5. *-commutative 58.1%

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

      6. cancel-sign-sub-inv 58.1%

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

      7. distribute-lft-neg-in 58.1%

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

      8. distribute-rgt-neg-in 58.1%

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

   6. Simplified 63.2%

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

      1. associate-*r/ 58.2%

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

      2. associate-/r/ 58.2%

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

      3. times-frac 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. times-frac 75.9%

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

      2. *-commutative 75.9%

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

   10. Applied egg-rr 75.9%

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

   11. Taylor expanded in D around 0 66.0%

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

       1. associate-/l/ 68.6%

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

       2. associate-*l/ 71.0%

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

       3. unpow2 71.0%

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

       4. associate-*r/ 73.5%

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

       5. *-commutative 73.5%

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

       6. associate-*r/ 73.7%

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

       7. associate-/l* 74.0%

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

       8. associate-/r/ 76.3%

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

   13. Simplified 76.3%

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

   if -2e103 < h

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in M around 0 51.9%

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

      1. *-commutative 51.9%

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

      2. *-commutative 51.9%

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

      3. times-frac 53.7%

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

      4. *-commutative 53.7%

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

      5. unpow2 53.7%

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

      6. unpow2 53.7%

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

      7. *-commutative 53.7%

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

      8. unpow2 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in D around 0 51.9%

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

      1. times-frac 52.8%

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

      2. unpow2 52.8%

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

      3. associate-*r/ 57.5%

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

      4. unpow2 57.5%

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

      5. *-commutative 57.5%

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

      6. associate-*l/ 58.9%

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

      7. unpow2 58.9%

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

      8. associate-*l* 64.7%

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

      9. *-commutative 64.7%

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

      10. associate-*r/ 67.7%

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

      11. *-commutative 67.7%

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

      12. associate-*l* 68.5%

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

      13. *-commutative 68.5%

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

      14. unpow2 68.5%

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

      15. associate-*r/ 66.9%

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

      16. unpow2 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in M around 0 66.2%

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

       1. associate-/l* 66.2%

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

       2. unpow2 66.2%

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

       3. *-commutative 66.2%

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

       4. associate-/r* 67.7%

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

       5. unpow2 67.7%

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

       6. times-frac 78.8%

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

       7. unpow2 78.8%

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

   12. Simplified 78.8%

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

       1. unpow2 78.8%

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

   14. Applied egg-rr 78.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+103}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{D}{\ell}\right)\right) \cdot \frac{h}{\frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 6: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D \cdot \frac{D}{\ell}}{\frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.8e-90)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ h (/ d M)) (/ (* D (/ D l)) (/ d M)))))))
   (* w0 (+ 1.0 (* (* D (/ (/ D (/ (* (/ d M) (/ d M)) h)) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.8e-90) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((h / (d / M)) * ((D * (D / l)) / (d / M))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 4.8d-90) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((h / (d_1 / m)) * ((d * (d / l)) / (d_1 / m))))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d / (((d_1 / m) * (d_1 / m)) / h)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.8e-90) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((h / (d / M)) * ((D * (D / l)) / (d / M))))));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.8e-90:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((h / (d / M)) * ((D * (D / l)) / (d / M))))))
	else:
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.8e-90)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(h / Float64(d / M)) * Float64(Float64(D * Float64(D / l)) / Float64(d / M)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D / Float64(Float64(Float64(d / M) * Float64(d / M)) / h)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.8e-90)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((h / (d / M)) * ((D * (D / l)) / (d / M))))));
	else
		tmp = w0 * (1.0 + ((D * ((D / (((d / M) * (d / M)) / h)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.8e-90], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.8000000000000003e-90

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. times-frac 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in w0 around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. cancel-sign-sub-inv 52.0%

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

      3. *-commutative 52.0%

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

      4. cancel-sign-sub-inv 52.0%

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

      5. *-commutative 52.0%

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

      6. cancel-sign-sub-inv 52.0%

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

      7. distribute-lft-neg-in 52.0%

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

      8. distribute-rgt-neg-in 52.0%

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

   6. Simplified 58.4%

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

      1. associate-*r/ 54.0%

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

      2. associate-/r/ 54.0%

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

      3. times-frac 70.9%

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

   8. Applied egg-rr 70.9%

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

      1. times-frac 79.8%

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

      2. *-commutative 79.8%

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

   10. Applied egg-rr 79.8%

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

   if 4.8000000000000003e-90 < d

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in M around 0 55.2%

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

      1. *-commutative 55.2%

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

      2. *-commutative 55.2%

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

      3. times-frac 56.2%

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

      4. *-commutative 56.2%

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

      5. unpow2 56.2%

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

      6. unpow2 56.2%

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

      7. *-commutative 56.2%

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

      8. unpow2 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in D around 0 55.2%

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

      1. times-frac 56.2%

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

      2. unpow2 56.2%

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

      3. associate-*r/ 59.3%

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

      4. unpow2 59.3%

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

      5. *-commutative 59.3%

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

      6. associate-*l/ 61.3%

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

      7. unpow2 61.3%

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

      8. associate-*l* 70.9%

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

      9. *-commutative 70.9%

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

      10. associate-*r/ 74.1%

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

      11. *-commutative 74.1%

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

      12. associate-*l* 76.0%

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

      13. *-commutative 76.0%

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

      14. unpow2 76.0%

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

      15. associate-*r/ 73.9%

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

      16. unpow2 73.9%

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

   9. Simplified 73.9%

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

   10. Taylor expanded in M around 0 72.0%

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

       1. associate-/l* 72.0%

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

       2. unpow2 72.0%

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

       3. *-commutative 72.0%

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

       4. associate-/r* 75.1%

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

       5. unpow2 75.1%

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

       6. times-frac 83.4%

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

       7. unpow2 83.4%

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

   12. Simplified 83.4%

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

       1. unpow2 83.4%

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

   14. Applied egg-rr 83.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D \cdot \frac{D}{\ell}}{\frac{d}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 7: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{M} \cdot \frac{d}{M}\\ \mathbf{if}\;d \leq 4 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot t_0} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{t_0}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (* (/ d M) (/ d M))))
   (if (<= d 4e-13)
     (* w0 (sqrt (+ 1.0 (* (/ (* D h) (* (/ l D) t_0)) -0.25))))
     (* w0 (+ 1.0 (* (* D (/ (/ D (/ t_0 h)) l)) -0.125))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (d / M) * (d / M);
	double tmp;
	if (d <= 4e-13) {
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * t_0)) * -0.25)));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (t_0 / h)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / m) * (d_1 / m)
    if (d_1 <= 4d-13) then
        tmp = w0 * sqrt((1.0d0 + (((d * h) / ((l / d) * t_0)) * (-0.25d0))))
    else
        tmp = w0 * (1.0d0 + ((d * ((d / (t_0 / h)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (d / M) * (d / M);
	double tmp;
	if (d <= 4e-13) {
		tmp = w0 * Math.sqrt((1.0 + (((D * h) / ((l / D) * t_0)) * -0.25)));
	} else {
		tmp = w0 * (1.0 + ((D * ((D / (t_0 / h)) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = (d / M) * (d / M)
	tmp = 0
	if d <= 4e-13:
		tmp = w0 * math.sqrt((1.0 + (((D * h) / ((l / D) * t_0)) * -0.25)))
	else:
		tmp = w0 * (1.0 + ((D * ((D / (t_0 / h)) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(d / M) * Float64(d / M))
	tmp = 0.0
	if (d <= 4e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D * h) / Float64(Float64(l / D) * t_0)) * -0.25))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D / Float64(t_0 / h)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (d / M) * (d / M);
	tmp = 0.0;
	if (d <= 4e-13)
		tmp = w0 * sqrt((1.0 + (((D * h) / ((l / D) * t_0)) * -0.25)));
	else
		tmp = w0 * (1.0 + ((D * ((D / (t_0 / h)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 4e-13], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D / N[(t$95$0 / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 4.0000000000000001e-13

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. times-frac 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in w0 around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. cancel-sign-sub-inv 52.5%

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

      3. *-commutative 52.5%

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

      4. cancel-sign-sub-inv 52.5%

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

      5. *-commutative 52.5%

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

      6. cancel-sign-sub-inv 52.5%

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

      7. distribute-lft-neg-in 52.5%

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

      8. distribute-rgt-neg-in 52.5%

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

   6. Simplified 59.0%

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

      1. frac-times 62.0%

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

      2. times-frac 78.2%

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

   8. Applied egg-rr 78.2%

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

   if 4.0000000000000001e-13 < d

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in M around 0 54.7%

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

      1. *-commutative 54.7%

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

      2. *-commutative 54.7%

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

      3. times-frac 55.8%

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

      4. *-commutative 55.8%

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

      5. unpow2 55.8%

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

      6. unpow2 55.8%

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

      7. *-commutative 55.8%

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

      8. unpow2 55.8%

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

   6. Simplified 55.8%

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

   7. Taylor expanded in D around 0 54.7%

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

      1. times-frac 55.8%

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

      2. unpow2 55.8%

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

      3. associate-*r/ 58.1%

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

      4. unpow2 58.1%

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

      5. *-commutative 58.1%

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

      6. associate-*l/ 61.6%

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

      7. unpow2 61.6%

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

      8. associate-*l* 71.3%

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

      9. *-commutative 71.3%

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

      10. associate-*r/ 74.9%

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

      11. *-commutative 74.9%

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

      12. associate-*l* 76.0%

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

      13. *-commutative 76.0%

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

      14. unpow2 76.0%

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

      15. associate-*r/ 72.4%

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

      16. unpow2 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in M around 0 71.4%

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

       1. associate-/l* 71.4%

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

       2. unpow2 71.4%

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

       3. *-commutative 71.4%

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

       4. associate-/r* 75.0%

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

       5. unpow2 75.0%

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

       6. times-frac 84.4%

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

       7. unpow2 84.4%

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

   12. Simplified 84.4%

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

       1. unpow2 84.4%

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

   14. Applied egg-rr 84.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{\frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 8: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-122}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e-122)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D (* d d)) (/ (* h (* M M)) l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e-122) {
		tmp = w0;
	} 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: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4d-122) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / (d_1 * d_1)) * ((h * (m * m)) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e-122) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / (d * d)) * ((h * (M * M)) / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e-122:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / (d * d)) * ((h * (M * M)) / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e-122)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e-122)
		tmp = w0;
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e-122], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000024e-122

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in M around 0 74.3%

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

   if 4.00000000000000024e-122 < M

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. times-frac 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 46.4%

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

      1. *-commutative 46.4%

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

      2. *-commutative 46.4%

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

      3. times-frac 49.4%

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

      4. *-commutative 49.4%

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

      5. unpow2 49.4%

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

      6. unpow2 49.4%

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

      7. *-commutative 49.4%

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

      8. unpow2 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in D around 0 46.4%

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

      1. times-frac 47.8%

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

      2. unpow2 47.8%

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

      3. associate-*r/ 54.9%

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

      4. unpow2 54.9%

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

      5. *-commutative 54.9%

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

      6. associate-*l/ 56.5%

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

      7. unpow2 56.5%

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

      8. associate-*l* 59.7%

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

      9. *-commutative 59.7%

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

      10. associate-*r/ 61.2%

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

      11. *-commutative 61.2%

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

      12. associate-*l* 62.6%

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

      13. *-commutative 62.6%

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

      14. unpow2 62.6%

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

      15. associate-*r/ 62.5%

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

      16. unpow2 62.5%

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

   9. Simplified 62.5%

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

   10. Taylor expanded in M around 0 56.7%

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

       1. unpow2 56.7%

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

       2. *-commutative 56.7%

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

       3. unpow2 56.7%

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

       4. *-commutative 56.7%

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

       5. times-frac 59.7%

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

       6. unpow2 59.7%

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

       7. unpow2 59.7%

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

       8. *-commutative 59.7%

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

   12. Simplified 59.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-122}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 9: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.6 \cdot 10^{-126}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot h}{d \cdot d}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 6.6e-126)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (/ (* (* M M) (/ (* D h) (* d d))) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.6e-126) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * M) * ((D * h) / (d * d))) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 6.6d-126) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * (((m * m) * ((d * h) / (d_1 * d_1))) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.6e-126) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * M) * ((D * h) / (d * d))) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6.6e-126:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * M) * ((D * h) / (d * d))) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 6.6e-126)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(Float64(M * M) * Float64(Float64(D * h) / Float64(d * d))) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 6.6e-126)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * M) * ((D * h) / (d * d))) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 6.6e-126], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(N[(M * M), $MachinePrecision] * N[(N[(D * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.6000000000000001e-126

   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 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in M around 0 74.2%

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

   if 6.6000000000000001e-126 < M

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. times-frac 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in M around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. *-commutative 47.1%

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

      3. times-frac 50.1%

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

      4. *-commutative 50.1%

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

      5. unpow2 50.1%

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

      6. unpow2 50.1%

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

      7. *-commutative 50.1%

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

      8. unpow2 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in D around 0 47.1%

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

      1. times-frac 48.5%

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

      2. unpow2 48.5%

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

      3. associate-*r/ 55.6%

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

      4. unpow2 55.6%

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

      5. *-commutative 55.6%

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

      6. associate-*l/ 57.1%

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

      7. unpow2 57.1%

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

      8. associate-*l* 60.3%

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

      9. *-commutative 60.3%

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

      10. associate-*r/ 61.8%

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

      11. *-commutative 61.8%

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

      12. associate-*l* 63.2%

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

      13. *-commutative 63.2%

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

      14. unpow2 63.2%

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

      15. associate-*r/ 63.0%

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

      16. unpow2 63.0%

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

   9. Simplified 63.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.6 \cdot 10^{-126}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{\left(M \cdot M\right) \cdot \frac{D \cdot h}{d \cdot d}}{\ell}\right)\right)\\ \end{array} \]

Alternative 10: 78.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{-153}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{\frac{M \cdot M}{d} \cdot \frac{D \cdot h}{d}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.5e-153)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (/ (* (/ (* M M) d) (/ (* D h) d)) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.5e-153) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((((M * M) / d) * ((D * h) / d)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.5d-153) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((((m * m) / d_1) * ((d * h) / d_1)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.5e-153) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((((M * M) / d) * ((D * h) / d)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.5e-153:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((((M * M) / d) * ((D * h) / d)) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.5e-153)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(Float64(Float64(M * M) / d) * Float64(Float64(D * h) / d)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.5e-153)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((((M * M) / d) * ((D * h) / d)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.5e-153], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.49999999999999981e-153

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 74.3%

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

   if 3.49999999999999981e-153 < M

   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 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in M around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. *-commutative 46.8%

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

      3. times-frac 49.6%

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

      4. *-commutative 49.6%

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

      5. unpow2 49.6%

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

      6. unpow2 49.6%

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

      7. *-commutative 49.6%

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

      8. unpow2 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in D around 0 46.8%

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

      1. times-frac 48.1%

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

      2. unpow2 48.1%

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

      3. associate-*r/ 54.5%

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

      4. unpow2 54.5%

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

      5. *-commutative 54.5%

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

      6. associate-*l/ 57.2%

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

      7. unpow2 57.2%

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

      8. associate-*l* 60.3%

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

      9. *-commutative 60.3%

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

      10. associate-*r/ 61.6%

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

      11. *-commutative 61.6%

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

      12. associate-*l* 62.9%

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

      13. *-commutative 62.9%

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

      14. unpow2 62.9%

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

      15. associate-*r/ 62.6%

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

      16. unpow2 62.6%

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

   9. Simplified 62.6%

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

   10. Taylor expanded in M around 0 58.8%

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

       1. *-commutative 58.8%

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

       2. unpow2 58.8%

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

       3. *-commutative 58.8%

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

       4. unpow2 58.8%

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

       5. unpow2 58.8%

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

       6. unpow2 58.8%

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

       7. associate-*r* 61.2%

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

       8. *-commutative 61.2%

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

       9. times-frac 67.9%

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

   12. Simplified 67.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{-153}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{\frac{M \cdot M}{d} \cdot \frac{D \cdot h}{d}}{\ell}\right)\right)\\ \end{array} \]

Alternative 11: 80.7% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((d * ((d / (((d_1 / m) * (d_1 / m)) / h)) / l)) * (-0.125d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. *-commutative 82.5%

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

   2. times-frac 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in M around 0 52.5%

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

   1. *-commutative 52.5%

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

   2. *-commutative 52.5%

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

   3. times-frac 54.1%

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

   4. *-commutative 54.1%

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

   5. unpow2 54.1%

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

   6. unpow2 54.1%

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

   7. *-commutative 54.1%

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

   8. unpow2 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in D around 0 52.5%

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

   1. times-frac 53.3%

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

   2. unpow2 53.3%

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

   3. associate-*r/ 57.6%

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

   4. unpow2 57.6%

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

   5. *-commutative 57.6%

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

   6. associate-*l/ 58.4%

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

   7. unpow2 58.4%

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

   8. associate-*l* 63.8%

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

   9. *-commutative 63.8%

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

   10. associate-*r/ 66.2%

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

   11. *-commutative 66.2%

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

   12. associate-*l* 66.9%

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

   13. *-commutative 66.9%

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

   14. unpow2 66.9%

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

   15. associate-*r/ 65.2%

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

   16. unpow2 65.2%

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

9. Simplified 65.2%

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

10. Taylor expanded in M around 0 65.4%

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

    1. associate-/l* 65.4%

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

    2. unpow2 65.4%

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

    3. *-commutative 65.4%

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

    4. associate-/r* 67.1%

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

    5. unpow2 67.1%

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

    6. times-frac 77.3%

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

    7. unpow2 77.3%

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

12. Simplified 77.3%

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

    1. unpow2 77.3%

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

14. Applied egg-rr 77.3%

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

15. Final simplification 77.3%

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

Alternative 12: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. *-commutative 82.5%

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

   2. times-frac 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in M around 0 70.8%

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

5. Final simplification 70.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023242 
(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))))))