Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 86.2%

Time: 18.7s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt[3]{{\left(1 - \frac{h \cdot {\left(\frac{0.5}{d} \cdot \left(M\_m \cdot D\_m\right)\right)}^{2}}{\ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M\_m}{d}}{\frac{2}{M\_m \cdot D\_m}} \cdot \left(h \cdot \frac{\frac{D\_m}{d}}{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 5e-163)
   (*
    w0
    (cbrt (pow (- 1.0 (/ (* h (pow (* (/ 0.5 d) (* M_m D_m)) 2.0)) l)) 1.5)))
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (/ (/ M_m d) (/ 2.0 (* M_m D_m))) (* h (/ (/ D_m d) 2.0))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5e-163) {
		tmp = w0 * cbrt(pow((1.0 - ((h * pow(((0.5 / d) * (M_m * D_m)), 2.0)) / l)), 1.5));
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5e-163) {
		tmp = w0 * Math.cbrt(Math.pow((1.0 - ((h * Math.pow(((0.5 / d) * (M_m * D_m)), 2.0)) / l)), 1.5));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 5e-163)
		tmp = Float64(w0 * cbrt((Float64(1.0 - Float64(Float64(h * (Float64(Float64(0.5 / d) * Float64(M_m * D_m)) ^ 2.0)) / l)) ^ 1.5)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) / Float64(2.0 / Float64(M_m * D_m))) * Float64(h * Float64(Float64(D_m / d) / 2.0))) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 5e-163], N[(w0 * N[Power[N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] / N[(2.0 / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999977e-163

   1. Initial program 83.3%

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

   3. Simplified 82.7%

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

      1. unpow2 82.7%

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

      2. unpow2 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*l/ 83.3%

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

      5. associate-*r/ 81.5%

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

      6. times-frac 83.3%

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

      7. associate-*r/ 90.3%

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

      8. *-commutative 90.3%

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

      9. associate-/l* 88.5%

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

      10. div-inv 88.5%

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

      11. associate-/r* 88.5%

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

      12. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. add-cbrt-cube 87.3%

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

      2. pow1/3 87.1%

         \[\leadsto w0 \cdot \color{blue}{{\left(\left(\sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}} \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\right) \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\right)}^{0.3333333333333333}} \]

   8. Applied egg-rr 88.2%

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - h \cdot \frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 88.5%

         \[\leadsto w0 \cdot \color{blue}{\sqrt[3]{{\left(1 - h \cdot \frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{1.5}}} \]

      2. associate-*r/ 89.1%

         \[\leadsto w0 \cdot \sqrt[3]{{\left(1 - \color{blue}{\frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\right)}^{1.5}} \]

      3. *-commutative 89.1%

         \[\leadsto w0 \cdot \sqrt[3]{{\left(1 - \frac{h \cdot {\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{1.5}} \]

   10. Simplified 89.1%

       \[\leadsto w0 \cdot \color{blue}{\sqrt[3]{{\left(1 - \frac{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{1.5}}} \]

   if 4.99999999999999977e-163 < M

   1. Initial program 71.4%

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

   3. Simplified 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*l/ 71.4%

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

      3. associate-*r/ 71.4%

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

      4. times-frac 71.4%

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

      5. unpow2 71.4%

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

      6. clear-num 71.4%

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

      7. frac-times 70.4%

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

      8. *-commutative 70.4%

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

      9. *-un-lft-identity 70.4%

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

      10. *-commutative 70.4%

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

      11. times-frac 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. *-commutative 70.4%

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

      2. times-frac 69.5%

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

   8. Simplified 69.5%

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

      1. associate-*r/ 75.8%

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

      2. frac-times 76.7%

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

      3. *-commutative 76.7%

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

      4. frac-times 76.7%

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

      5. *-commutative 76.7%

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

      6. *-commutative 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. times-frac 76.9%

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

       2. times-frac 75.8%

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

   12. Applied egg-rr 75.8%

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

       1. associate-/r* 75.8%

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

   14. Simplified 75.8%

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

       1. pow1 75.8%

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

       2. associate-*l* 79.7%

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

       3. frac-times 81.7%

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

       4. associate-/l* 81.7%

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

       5. associate-/l/ 81.7%

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

       6. *-commutative 81.7%

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

   16. Applied egg-rr 81.7%

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

       1. unpow1 81.7%

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

       2. associate-/r* 79.7%

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

       3. *-commutative 79.7%

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

       4. *-commutative 79.7%

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

       5. associate-/r* 79.7%

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

   18. Simplified 79.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt[3]{{\left(1 - \frac{h \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}}{\ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{d}}{\frac{2}{M \cdot D}} \cdot \left(h \cdot \frac{\frac{D}{d}}{2}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M\_m}{d}}{\frac{2}{M\_m \cdot D\_m}} \cdot \left(h \cdot \frac{\frac{D\_m}{d}}{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ h l)))))
   (if (<= t_0 5e+201)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (* (/ (/ M_m d) (/ 2.0 (* M_m D_m))) (* h (/ (/ D_m d) 2.0)))
         l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+201) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+201) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((((m_m / d) / (2.0d0 / (m_m * d_m))) * (h * ((d_m / d) / 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+201) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 1.0 - (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+201:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+201)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) / Float64(2.0 / Float64(M_m * D_m))) * Float64(h * Float64(Float64(D_m / d) / 2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 1.0 - ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+201)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+201], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] / N[(2.0 / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 4.9999999999999995e201

   1. Initial program 99.9%

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

   if 4.9999999999999995e201 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 38.9%

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

   3. Simplified 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*l/ 38.9%

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

      3. associate-*r/ 38.9%

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

      4. times-frac 38.9%

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

      5. unpow2 38.9%

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

      6. clear-num 38.9%

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

      7. frac-times 38.9%

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

      8. *-commutative 38.9%

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

      9. *-un-lft-identity 38.9%

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

      10. *-commutative 38.9%

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

      11. times-frac 38.8%

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

   6. Applied egg-rr 38.8%

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

      1. *-commutative 38.8%

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

      2. times-frac 38.8%

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

   8. Simplified 38.8%

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

      1. associate-*r/ 57.3%

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

      2. frac-times 58.3%

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

      3. *-commutative 58.3%

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

      4. frac-times 58.4%

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

      5. *-commutative 58.4%

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

      6. *-commutative 58.4%

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

   10. Applied egg-rr 58.4%

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

       1. times-frac 57.4%

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

       2. times-frac 57.4%

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

   12. Applied egg-rr 57.4%

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

       1. associate-/r* 57.4%

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

   14. Simplified 57.4%

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

       1. pow1 57.4%

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

       2. associate-*l* 62.8%

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

       3. frac-times 63.9%

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

       4. associate-/l* 63.9%

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

       5. associate-/l/ 63.9%

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

       6. *-commutative 63.9%

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

   16. Applied egg-rr 63.9%

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

       1. unpow1 63.9%

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

       2. associate-/r* 62.7%

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

       3. *-commutative 62.7%

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

       4. *-commutative 62.7%

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

       5. associate-/r* 62.7%

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

   18. Simplified 62.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{d}}{\frac{2}{M \cdot D}} \cdot \left(h \cdot \frac{\frac{D}{d}}{2}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}\right) \cdot \left(\frac{\frac{D\_m}{d}}{2} \cdot \left(M\_m \cdot D\_m\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (/ 0.5 d)))))
   (if (<= (/ h l) (- INFINITY))
     (*
      w0
      (sqrt
       (-
        1.0
        (* (* M_m (/ h (* d (* 2.0 l)))) (* (/ (/ D_m d) 2.0) (* M_m D_m))))))
     (if (<= (/ h l) -2e-310)
       (* w0 (sqrt (- 1.0 (* (/ h l) (* t_0 t_0)))))
       w0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (D_m * (0.5 / d));
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	} else if ((h / l) <= -2e-310) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (D_m * (0.5 / d));
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	} else if ((h / l) <= -2e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = M_m * (D_m * (0.5 / d))
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))))
	elif (h / l) <= -2e-310:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (t_0 * t_0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(D_m * Float64(0.5 / d)))
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(h / Float64(d * Float64(2.0 * l)))) * Float64(Float64(Float64(D_m / d) / 2.0) * Float64(M_m * D_m))))));
	elseif (Float64(h / l) <= -2e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = M_m * (D_m * (0.5 / d));
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	elseif ((h / l) <= -2e-310)
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(h / N[(d * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 28.8%

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

   3. Simplified 28.7%

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

      1. *-commutative 28.7%

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

      2. associate-*l/ 28.8%

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

      3. associate-*r/ 28.8%

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

      4. times-frac 28.8%

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

      5. unpow2 28.8%

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

      6. clear-num 28.8%

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

      7. frac-times 28.8%

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

      8. *-commutative 28.8%

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

      9. *-un-lft-identity 28.8%

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

      10. *-commutative 28.8%

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

      11. times-frac 28.8%

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

   6. Applied egg-rr 28.8%

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

      1. *-commutative 28.8%

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

      2. times-frac 28.7%

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

   8. Simplified 28.7%

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

      1. pow1/2 28.7%

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

      2. associate-*l* 28.7%

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

      3. frac-times 28.9%

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

      4. *-commutative 28.9%

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

      5. *-commutative 28.9%

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

   10. Applied egg-rr 28.9%

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

       1. *-un-lft-identity 28.9%

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

       2. unpow1/2 28.9%

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

       3. associate-/r/ 28.9%

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

       4. *-commutative 28.9%

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

   12. Applied egg-rr 28.9%

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

       1. *-lft-identity 28.9%

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

       2. *-commutative 28.9%

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

       3. *-commutative 28.9%

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

       4. associate-/r* 28.9%

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

       5. *-commutative 28.9%

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

       6. times-frac 73.0%

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

       7. associate-/l* 67.4%

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

       8. associate-*l* 67.4%

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

   14. Simplified 67.4%

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

   if -inf.0 < (/.f64 h l) < -1.999999999999994e-310

   1. Initial program 81.0%

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

   3. Simplified 80.2%

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

      1. metadata-eval 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*l/ 81.0%

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

      4. associate-*r/ 79.6%

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

      5. times-frac 81.0%

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

      6. pow-prod-up 81.0%

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

      7. associate-/l* 79.6%

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

      8. pow1 79.6%

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

      9. div-inv 79.6%

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

      10. associate-/r* 79.6%

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

      11. metadata-eval 79.6%

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

      12. associate-/l* 79.6%

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

      13. pow1 79.6%

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

      14. div-inv 79.6%

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

      15. associate-/r* 79.6%

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

      16. metadata-eval 79.6%

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

   6. Applied egg-rr 79.6%

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

   if -1.999999999999994e-310 < (/.f64 h l)

   1. Initial program 84.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in D around 0 93.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot D\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-132}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}\right) \cdot \left(\frac{\frac{D\_m}{d}}{2} \cdot \left(M\_m \cdot D\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e-132)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (/ h (* d (* 2.0 l)))) (* (/ (/ D_m d) 2.0) (* M_m D_m))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-132) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 2d-132) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * (h / (d * (2.0d0 * l)))) * (((d_m / d) / 2.0d0) * (m_m * d_m)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-132) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 2e-132:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-132)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(h / Float64(d * Float64(2.0 * l)))) * Float64(Float64(Float64(D_m / d) / 2.0) * Float64(M_m * D_m))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 2e-132)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m * (h / (d * (2.0 * l)))) * (((D_m / d) / 2.0) * (M_m * D_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-132], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(h / N[(d * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2e-132

   1. Initial program 83.2%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 72.5%

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

   if 2e-132 < (*.f64 M D)

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*l/ 67.1%

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

      3. associate-*r/ 65.8%

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

      4. times-frac 67.1%

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

      5. unpow2 67.1%

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

      6. clear-num 67.1%

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

      7. frac-times 67.1%

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

      8. *-commutative 67.1%

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

      9. *-un-lft-identity 67.1%

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

      10. *-commutative 67.1%

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

      11. times-frac 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. *-commutative 65.8%

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

      2. times-frac 63.1%

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

   8. Simplified 63.1%

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

      1. pow1/2 63.2%

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

      2. associate-*l* 64.3%

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

      3. frac-times 64.4%

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

      4. *-commutative 64.4%

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

      5. *-commutative 64.4%

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

   10. Applied egg-rr 64.4%

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

       1. *-un-lft-identity 64.4%

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

       2. unpow1/2 64.4%

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

       3. associate-/r/ 64.4%

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

       4. *-commutative 64.4%

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

   12. Applied egg-rr 64.4%

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

       1. *-lft-identity 64.4%

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

       2. *-commutative 64.4%

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

       3. *-commutative 64.4%

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

       4. associate-/r* 64.4%

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

       5. *-commutative 64.4%

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

       6. times-frac 72.7%

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

       7. associate-/l* 71.2%

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

       8. associate-*l* 71.2%

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

   14. Simplified 71.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-132}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot D\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M\_m}{d}}{\frac{2}{M\_m \cdot D\_m}} \cdot \left(h \cdot \frac{\frac{D\_m}{d}}{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - D\_m \cdot \frac{M\_m \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}}{2 \cdot \frac{d}{M\_m \cdot D\_m}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 2e+72)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (/ (/ M_m d) (/ 2.0 (* M_m D_m))) (* h (/ (/ D_m d) 2.0))) l))))
   (*
    w0
    (pow
     (-
      1.0
      (* D_m (/ (* M_m (/ h (* d (* 2.0 l)))) (* 2.0 (/ d (* M_m D_m))))))
     0.5))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2e+72) {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	} else {
		tmp = w0 * pow((1.0 - (D_m * ((M_m * (h / (d * (2.0 * l)))) / (2.0 * (d / (M_m * D_m)))))), 0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 2d+72) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m / d) / (2.0d0 / (m_m * d_m))) * (h * ((d_m / d) / 2.0d0))) / l)))
    else
        tmp = w0 * ((1.0d0 - (d_m * ((m_m * (h / (d * (2.0d0 * l)))) / (2.0d0 * (d / (m_m * d_m)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2e+72) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	} else {
		tmp = w0 * Math.pow((1.0 - (D_m * ((M_m * (h / (d * (2.0 * l)))) / (2.0 * (d / (M_m * D_m)))))), 0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 2e+72:
		tmp = w0 * math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)))
	else:
		tmp = w0 * math.pow((1.0 - (D_m * ((M_m * (h / (d * (2.0 * l)))) / (2.0 * (d / (M_m * D_m)))))), 0.5)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 2e+72)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) / Float64(2.0 / Float64(M_m * D_m))) * Float64(h * Float64(Float64(D_m / d) / 2.0))) / l))));
	else
		tmp = Float64(w0 * (Float64(1.0 - Float64(D_m * Float64(Float64(M_m * Float64(h / Float64(d * Float64(2.0 * l)))) / Float64(2.0 * Float64(d / Float64(M_m * D_m)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 2e+72)
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	else
		tmp = w0 * ((1.0 - (D_m * ((M_m * (h / (d * (2.0 * l)))) / (2.0 * (d / (M_m * D_m)))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 2e+72], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] / N[(2.0 / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 - N[(D$95$m * N[(N[(M$95$m * N[(h / N[(d * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.99999999999999989e72

   1. Initial program 82.2%

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

   3. Simplified 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*l/ 82.2%

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

      3. associate-*r/ 81.3%

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

      4. times-frac 82.2%

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

      5. unpow2 82.2%

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

      6. clear-num 82.2%

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

      7. frac-times 81.1%

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

      8. *-commutative 81.1%

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

      9. *-un-lft-identity 81.1%

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

      10. *-commutative 81.1%

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

      11. times-frac 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. times-frac 80.3%

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

   8. Simplified 80.3%

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

      1. associate-*r/ 87.3%

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

      2. frac-times 87.1%

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

      3. *-commutative 87.1%

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

      4. frac-times 88.0%

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

      5. *-commutative 88.0%

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

      6. *-commutative 88.0%

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

   10. Applied egg-rr 88.0%

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

       1. times-frac 87.4%

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

       2. times-frac 86.4%

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

   12. Applied egg-rr 86.4%

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

       1. associate-/r* 86.4%

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

   14. Simplified 86.4%

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

       1. pow1 86.4%

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

       2. associate-*l* 87.3%

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

       3. frac-times 88.7%

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

       4. associate-/l* 88.7%

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

       5. associate-/l/ 88.7%

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

       6. *-commutative 88.7%

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

   16. Applied egg-rr 88.7%

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

       1. unpow1 88.7%

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

       2. associate-/r* 87.3%

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

       3. *-commutative 87.3%

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

       4. *-commutative 87.3%

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

       5. associate-/r* 87.3%

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

   18. Simplified 87.3%

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

   if 1.99999999999999989e72 < D

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*l/ 64.8%

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

      3. associate-*r/ 63.0%

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

      4. times-frac 64.8%

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

      5. unpow2 64.8%

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

      6. clear-num 64.9%

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

      7. frac-times 63.0%

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

      8. *-commutative 63.0%

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

      9. *-un-lft-identity 63.0%

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

      10. *-commutative 63.0%

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

      11. times-frac 63.0%

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

   6. Applied egg-rr 63.0%

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

      1. *-commutative 63.0%

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

      2. times-frac 59.1%

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

   8. Simplified 59.1%

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

      1. pow1/2 59.3%

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

      2. associate-*l* 60.9%

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

      3. frac-times 62.8%

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

      4. *-commutative 62.8%

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

      5. *-commutative 62.8%

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

   10. Applied egg-rr 62.8%

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

       1. frac-times 62.8%

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

       2. associate-*l/ 69.9%

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

       3. *-commutative 69.9%

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

       4. frac-times 68.6%

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

       5. associate-/l* 68.6%

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

   12. Applied egg-rr 68.6%

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

       1. associate-/l* 68.6%

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

       2. *-commutative 68.6%

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

       3. times-frac 74.4%

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

       4. associate-/l* 74.1%

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

       5. associate-*l* 74.1%

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

       6. associate-*r/ 74.1%

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

       7. *-commutative 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-*r/ 74.1%

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

   14. Simplified 74.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{d}}{\frac{2}{M \cdot D}} \cdot \left(h \cdot \frac{\frac{D}{d}}{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - D \cdot \frac{M \cdot \frac{h}{d \cdot \left(2 \cdot \ell\right)}}{2 \cdot \frac{d}{M \cdot D}}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M\_m \cdot D\_m}{\left(d \cdot 2\right) \cdot \frac{d \cdot 2}{M\_m \cdot D\_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M\_m}{d}}{\frac{2}{M\_m \cdot D\_m}} \cdot \left(h \cdot \frac{\frac{D\_m}{d}}{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 4e-163)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* h (/ (* M_m D_m) (* (* d 2.0) (/ (* d 2.0) (* M_m D_m))))) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (/ (/ M_m d) (/ 2.0 (* M_m D_m))) (* h (/ (/ D_m d) 2.0))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e-163) {
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 4d-163) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * d_m) / ((d * 2.0d0) * ((d * 2.0d0) / (m_m * d_m))))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((m_m / d) / (2.0d0 / (m_m * d_m))) * (h * ((d_m / d) / 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e-163) {
		tmp = w0 * Math.sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 4e-163:
		tmp = w0 * math.sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 4e-163)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(M_m * D_m) / Float64(Float64(d * 2.0) * Float64(Float64(d * 2.0) / Float64(M_m * D_m))))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) / Float64(2.0 / Float64(M_m * D_m))) * Float64(h * Float64(Float64(D_m / d) / 2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 4e-163)
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((((M_m / d) / (2.0 / (M_m * D_m))) * (h * ((D_m / d) / 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 4e-163], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] / N[(2.0 / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.99999999999999969e-163

   1. Initial program 83.3%

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

   3. Simplified 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l/ 83.3%

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

      3. associate-*r/ 81.5%

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

      4. times-frac 83.3%

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

      5. unpow2 83.3%

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

      6. clear-num 83.3%

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

      7. frac-times 81.8%

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

      8. *-commutative 81.8%

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

      9. *-un-lft-identity 81.8%

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

      10. *-commutative 81.8%

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

      11. times-frac 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. times-frac 80.2%

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

   8. Simplified 80.2%

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

      1. associate-*r/ 86.7%

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

      2. frac-times 87.6%

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

      3. *-commutative 87.6%

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

      4. frac-times 88.8%

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

      5. *-commutative 88.8%

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

      6. *-commutative 88.8%

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

   10. Applied egg-rr 88.8%

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

   if 3.99999999999999969e-163 < M

   1. Initial program 71.4%

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

   3. Simplified 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*l/ 71.4%

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

      3. associate-*r/ 71.4%

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

      4. times-frac 71.4%

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

      5. unpow2 71.4%

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

      6. clear-num 71.4%

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

      7. frac-times 70.4%

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

      8. *-commutative 70.4%

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

      9. *-un-lft-identity 70.4%

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

      10. *-commutative 70.4%

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

      11. times-frac 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. *-commutative 70.4%

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

      2. times-frac 69.5%

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

   8. Simplified 69.5%

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

      1. associate-*r/ 75.8%

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

      2. frac-times 76.7%

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

      3. *-commutative 76.7%

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

      4. frac-times 76.7%

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

      5. *-commutative 76.7%

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

      6. *-commutative 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. times-frac 76.9%

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

       2. times-frac 75.8%

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

   12. Applied egg-rr 75.8%

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

       1. associate-/r* 75.8%

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

   14. Simplified 75.8%

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

       1. pow1 75.8%

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

       2. associate-*l* 79.7%

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

       3. frac-times 81.7%

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

       4. associate-/l* 81.7%

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

       5. associate-/l/ 81.7%

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

       6. *-commutative 81.7%

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

   16. Applied egg-rr 81.7%

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

       1. unpow1 81.7%

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

       2. associate-/r* 79.7%

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

       3. *-commutative 79.7%

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

       4. *-commutative 79.7%

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

       5. associate-/r* 79.7%

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

   18. Simplified 79.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M \cdot D}{\left(d \cdot 2\right) \cdot \frac{d \cdot 2}{M \cdot D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M}{d}}{\frac{2}{M \cdot D}} \cdot \left(h \cdot \frac{\frac{D}{d}}{2}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+223}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M\_m \cdot D\_m}{\left(d \cdot 2\right) \cdot \frac{d \cdot 2}{M\_m \cdot D\_m}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= l -4e+223)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* h (/ (* M_m D_m) (* (* d 2.0) (/ (* d 2.0) (* M_m D_m))))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= -4e+223) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (l <= (-4d+223)) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * d_m) / ((d * 2.0d0) * ((d * 2.0d0) / (m_m * d_m))))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= -4e+223) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if l <= -4e+223:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (l <= -4e+223)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(M_m * D_m) / Float64(Float64(d * 2.0) * Float64(Float64(d * 2.0) / Float64(M_m * D_m))))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (l <= -4e+223)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * D_m) / ((d * 2.0) * ((d * 2.0) / (M_m * D_m))))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[l, -4e+223], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.00000000000000019e223

   1. Initial program 74.1%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in D around 0 88.2%

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

   if -4.00000000000000019e223 < l

   1. Initial program 79.2%

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

   3. Simplified 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*l/ 79.2%

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

      3. associate-*r/ 78.0%

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

      4. times-frac 79.2%

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

      5. unpow2 79.2%

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

      6. clear-num 79.3%

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

      7. frac-times 78.2%

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

      8. *-commutative 78.2%

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

      9. *-un-lft-identity 78.2%

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

      10. *-commutative 78.2%

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

      11. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. *-commutative 77.4%

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

      2. times-frac 76.7%

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

   8. Simplified 76.7%

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

      1. associate-*r/ 83.8%

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

      2. frac-times 84.9%

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

      3. *-commutative 84.9%

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

      4. frac-times 85.7%

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

      5. *-commutative 85.7%

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

      6. *-commutative 85.7%

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

   10. Applied egg-rr 85.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+223}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M \cdot D}{\left(d \cdot 2\right) \cdot \frac{d \cdot 2}{M \cdot D}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+232}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\frac{\frac{D\_m}{d}}{2} \cdot \frac{M\_m}{\frac{d}{M\_m} \cdot \frac{2}{D\_m}}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= l -3.2e+232)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* h (* (/ (/ D_m d) 2.0) (/ M_m (* (/ d M_m) (/ 2.0 D_m))))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= -3.2e+232) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h * (((D_m / d) / 2.0) * (M_m / ((d / M_m) * (2.0 / D_m))))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (l <= (-3.2d+232)) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((d_m / d) / 2.0d0) * (m_m / ((d / m_m) * (2.0d0 / d_m))))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= -3.2e+232) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * (((D_m / d) / 2.0) * (M_m / ((d / M_m) * (2.0 / D_m))))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if l <= -3.2e+232:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * (((D_m / d) / 2.0) * (M_m / ((d / M_m) * (2.0 / D_m))))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (l <= -3.2e+232)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(Float64(D_m / d) / 2.0) * Float64(M_m / Float64(Float64(d / M_m) * Float64(2.0 / D_m))))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (l <= -3.2e+232)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h * (((D_m / d) / 2.0) * (M_m / ((d / M_m) * (2.0 / D_m))))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[l, -3.2e+232], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * N[(M$95$m / N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.2000000000000002e232

   1. Initial program 74.1%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in D around 0 88.2%

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

   if -3.2000000000000002e232 < l

   1. Initial program 79.2%

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

   3. Simplified 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*l/ 79.2%

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

      3. associate-*r/ 78.0%

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

      4. times-frac 79.2%

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

      5. unpow2 79.2%

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

      6. clear-num 79.3%

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

      7. frac-times 78.2%

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

      8. *-commutative 78.2%

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

      9. *-un-lft-identity 78.2%

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

      10. *-commutative 78.2%

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

      11. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. *-commutative 77.4%

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

      2. times-frac 76.7%

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

   8. Simplified 76.7%

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

      1. associate-*r/ 83.8%

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

      2. frac-times 84.9%

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

      3. *-commutative 84.9%

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

      4. frac-times 85.7%

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

      5. *-commutative 85.7%

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

      6. *-commutative 85.7%

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

   10. Applied egg-rr 85.7%

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

       1. times-frac 84.3%

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

       2. times-frac 83.9%

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

   12. Applied egg-rr 83.9%

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

       1. associate-/r* 83.9%

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

   14. Simplified 83.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+232}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\frac{\frac{D}{d}}{2} \cdot \frac{M}{\frac{d}{M} \cdot \frac{2}{D}}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+279}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 5e+279)
   w0
   (* -0.125 (/ (* (* (* M_m D_m) (* M_m D_m)) (* w0 h)) (* l (pow d 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e+279) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M_m * D_m) * (M_m * D_m)) * (w0 * h)) / (l * pow(d, 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 5d+279) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((m_m * d_m) * (m_m * d_m)) * (w0 * h)) / (l * (d ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e+279) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M_m * D_m) * (M_m * D_m)) * (w0 * h)) / (l * Math.pow(d, 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 5e+279:
		tmp = w0
	else:
		tmp = -0.125 * ((((M_m * D_m) * (M_m * D_m)) * (w0 * h)) / (l * math.pow(d, 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e+279)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(w0 * h)) / Float64(l * (d ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 5e+279)
		tmp = w0;
	else
		tmp = -0.125 * ((((M_m * D_m) * (M_m * D_m)) * (w0 * h)) / (l * (d ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+279], w0, N[(-0.125 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000002e279

   1. Initial program 80.2%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in D around 0 72.2%

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

   if 5.0000000000000002e279 < (*.f64 M D)

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in D around 0 67.2%

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

      1. +-commutative 67.2%

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

      2. fma-define 67.2%

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

      3. associate-*r* 67.2%

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

      4. unpow2 67.2%

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

      5. unpow2 67.2%

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

      6. swap-sqr 67.2%

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

      7. unpow2 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in D around inf 67.2%

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

      1. associate-*r* 67.2%

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

      2. *-commutative 67.2%

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

      3. unpow2 67.2%

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

      4. unpow2 67.2%

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

      5. swap-sqr 67.2%

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

      6. unpow2 67.2%

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

      7. *-commutative 67.2%

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

      8. *-commutative 67.2%

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

   10. Simplified 67.2%

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

       1. unpow2 67.2%

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

       2. *-commutative 67.2%

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

       3. *-commutative 67.2%

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

   12. Applied egg-rr 67.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{+279}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 216.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d) :precision binary64 w0)

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 78.7%

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

3. Simplified 77.9%

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

5. Taylor expanded in D around 0 68.5%

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

Reproduce

herbie shell --seed 2024136 
(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))))))