Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.9% → 90.1%

Time: 16.0s

Alternatives: 6

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{-0.25 \cdot \frac{h \cdot {\left(M \cdot D\_m\right)}^{2}}{\ell}}}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(D\_m \cdot 0.5\right) \cdot \frac{M}{d\_m}\right) \cdot \frac{D\_m \cdot \left(0.5 \cdot \frac{M}{d\_m}\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+247)
   (* w0 (/ (sqrt (* -0.25 (/ (* h (pow (* M D_m) 2.0)) l))) d_m))
   (*
    w0
    (sqrt
     (-
      1.0
      (* h (* (* (* D_m 0.5) (/ M d_m)) (/ (* D_m (* 0.5 (/ M d_m))) l))))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+247) {
		tmp = w0 * (sqrt((-0.25 * ((h * pow((M * D_m), 2.0)) / l))) / d_m);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-1d+247)) then
        tmp = w0 * (sqrt(((-0.25d0) * ((h * ((m * d_m) ** 2.0d0)) / l))) / d_m_1)
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((d_m * 0.5d0) * (m / d_m_1)) * ((d_m * (0.5d0 * (m / d_m_1))) / l)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+247) {
		tmp = w0 * (Math.sqrt((-0.25 * ((h * Math.pow((M * D_m), 2.0)) / l))) / d_m);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+247:
		tmp = w0 * (math.sqrt((-0.25 * ((h * math.pow((M * D_m), 2.0)) / l))) / d_m)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+247)
		tmp = Float64(w0 * Float64(sqrt(Float64(-0.25 * Float64(Float64(h * (Float64(M * D_m) ^ 2.0)) / l))) / d_m));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(D_m * 0.5) * Float64(M / d_m)) * Float64(Float64(D_m * Float64(0.5 * Float64(M / d_m))) / l))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -1e+247)
		tmp = w0 * (sqrt((-0.25 * ((h * ((M * D_m) ^ 2.0)) / l))) / d_m);
	else
		tmp = w0 * sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+247], N[(w0 * N[(N[Sqrt[N[(-0.25 * N[(N[(h * N[Power[N[(M * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(0.5 * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999952e246

   1. Initial program 58.7%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in D around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. times-frac 40.1%

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

      3. associate-*r* 40.2%

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

   7. Simplified 40.2%

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

      1. associate-*l/ 40.2%

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

      2. sqrt-div 43.8%

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

      3. associate-/l* 45.1%

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

      4. pow-prod-down 59.7%

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

      5. sqrt-pow1 28.1%

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

      6. metadata-eval 28.1%

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

      7. pow1 28.1%

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

   9. Applied egg-rr 28.1%

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

       1. associate-*r/ 30.3%

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

   11. Applied egg-rr 30.3%

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

   if -9.99999999999999952e246 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 83.0%

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

   3. Simplified 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. unpow1 83.1%

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

      2. associate-*l/ 94.3%

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

      3. associate-/l* 94.3%

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

      4. associate-*r/ 94.3%

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

      5. times-frac 94.3%

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

      6. associate-*r/ 94.3%

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

   8. Simplified 94.3%

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

      1. div-inv 94.2%

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

      2. div-inv 94.2%

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

      3. metadata-eval 94.2%

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

   10. Applied egg-rr 94.2%

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

       1. unpow2 94.2%

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

       2. *-un-lft-identity 94.2%

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

       3. times-frac 96.3%

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

       4. associate-*l* 94.8%

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

       5. div-inv 94.8%

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

       6. associate-*l* 94.8%

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

       7. associate-*l* 96.2%

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

       8. div-inv 96.2%

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

       9. associate-*l* 96.2%

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

   12. Applied egg-rr 96.2%

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

       1. /-rgt-identity 96.2%

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

       2. associate-*r* 96.2%

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

   14. Applied egg-rr 96.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 1.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (* h (* (* (* D_m 0.5) (/ M d_m)) (/ (* D_m (* 0.5 (/ M d_m))) l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))));
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (h * (((d_m * 0.5d0) * (m / d_m_1)) * ((d_m * (0.5d0 * (m / d_m_1))) / l)))))
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (h * (((D_m * 0.5) * (M / d_m)) * ((D_m * (0.5 * (M / d_m))) / l)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(0.5 * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

6. Applied egg-rr 75.6%

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

   1. unpow1 75.6%

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

   2. associate-*l/ 83.8%

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

   3. associate-/l* 83.8%

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

   4. associate-*r/ 83.4%

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

   5. times-frac 83.8%

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

   6. associate-*r/ 83.4%

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

8. Simplified 83.4%

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

   1. div-inv 83.4%

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

   2. div-inv 83.4%

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

   3. metadata-eval 83.4%

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

10. Applied egg-rr 83.4%

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

    1. unpow2 83.4%

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

    2. *-un-lft-identity 83.4%

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

    3. times-frac 85.6%

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

    4. associate-*l* 84.2%

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

    5. div-inv 84.2%

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

    6. associate-*l* 84.2%

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

    7. associate-*l* 85.9%

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

    8. div-inv 85.9%

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

    9. associate-*l* 85.9%

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

12. Applied egg-rr 85.9%

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

    1. /-rgt-identity 85.9%

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

    2. associate-*r* 85.9%

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

14. Applied egg-rr 85.9%

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

Alternative 3: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d\_m}\\ w0 \cdot \sqrt{1 - h \cdot \frac{\frac{t\_0 \cdot t\_0}{4}}{\ell}} \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M d_m))))
   (* w0 (sqrt (- 1.0 (* h (/ (/ (* t_0 t_0) 4.0) l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M / d_m);
	return w0 * sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = d_m * (m / d_m_1)
    code = w0 * sqrt((1.0d0 - (h * (((t_0 * t_0) / 4.0d0) / l))))
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M / d_m);
	return w0 * Math.sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	t_0 = D_m * (M / d_m)
	return w0 * math.sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))))

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M / d_m))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(t_0 * t_0) / 4.0) / l)))))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp = code(w0, M, D_m, h, l, d_m)
	t_0 = D_m * (M / d_m);
	tmp = w0 * sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

6. Applied egg-rr 75.6%

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

   1. unpow1 75.6%

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

   2. associate-*l/ 83.8%

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

   3. associate-/l* 83.8%

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

   4. associate-*r/ 83.4%

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

   5. times-frac 83.8%

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

   6. associate-*r/ 83.4%

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

8. Simplified 83.4%

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

   1. div-inv 83.4%

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

   2. div-inv 83.4%

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

   3. metadata-eval 83.4%

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

10. Applied egg-rr 83.4%

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

    1. associate-*l* 83.8%

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

    2. div-inv 83.8%

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

    3. metadata-eval 83.8%

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

    4. div-inv 83.8%

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

    5. unpow2 83.8%

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

    6. associate-*l/ 83.8%

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

    7. associate-*l/ 83.8%

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

    8. frac-times 83.8%

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

    9. metadata-eval 83.8%

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

12. Applied egg-rr 83.8%

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

Alternative 4: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{\left(M \cdot D\_m\right) \cdot \frac{\frac{M \cdot D\_m}{d\_m}}{d\_m \cdot 4}}{\ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* h (/ (* (* M D_m) (/ (/ (* M D_m) d_m) (* d_m 4.0))) l))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (h * (((M * D_m) * (((M * D_m) / d_m) / (d_m * 4.0))) / l))));
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (h * (((m * d_m) * (((m * d_m) / d_m_1) / (d_m_1 * 4.0d0))) / l))))
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (h * (((M * D_m) * (((M * D_m) / d_m) / (d_m * 4.0))) / l))));
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h * (((M * D_m) * (((M * D_m) / d_m) / (d_m * 4.0))) / l))))

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D_m) * Float64(Float64(Float64(M * D_m) / d_m) / Float64(d_m * 4.0))) / l)))))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp = code(w0, M, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h * (((M * D_m) * (((M * D_m) / d_m) / (d_m * 4.0))) / l))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(N[(M * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] / N[(d$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

6. Applied egg-rr 75.6%

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

   1. unpow1 75.6%

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

   2. associate-*l/ 83.8%

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

   3. associate-/l* 83.8%

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

   4. associate-*r/ 83.4%

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

   5. times-frac 83.8%

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

   6. associate-*r/ 83.4%

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

8. Simplified 83.4%

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

   1. div-inv 83.4%

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

   2. div-inv 83.4%

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

   3. metadata-eval 83.4%

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

10. Applied egg-rr 83.4%

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

    1. associate-*l* 83.8%

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

    2. div-inv 83.8%

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

    3. metadata-eval 83.8%

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

    4. div-inv 83.8%

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

    5. unpow2 83.8%

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

    6. frac-times 83.0%

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

    7. associate-*l/ 83.0%

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

    8. frac-times 82.7%

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

12. Applied egg-rr 82.7%

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

    1. associate-/l* 81.1%

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

    2. *-commutative 81.1%

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

    3. associate-*r* 81.1%

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

    4. metadata-eval 81.1%

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

14. Simplified 81.1%

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

15. Taylor expanded in D around 0 81.5%

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

16. Final simplification 81.5%

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

Alternative 5: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{\left(M \cdot D\_m\right) \cdot \frac{D\_m \cdot \frac{M}{d\_m}}{d\_m \cdot 4}}{\ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* h (/ (* (* M D_m) (/ (* D_m (/ M d_m)) (* d_m 4.0))) l))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (h * (((M * D_m) * ((D_m * (M / d_m)) / (d_m * 4.0))) / l))));
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (h * (((m * d_m) * ((d_m * (m / d_m_1)) / (d_m_1 * 4.0d0))) / l))))
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (h * (((M * D_m) * ((D_m * (M / d_m)) / (d_m * 4.0))) / l))));
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h * (((M * D_m) * ((D_m * (M / d_m)) / (d_m * 4.0))) / l))))

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M * D_m) * Float64(Float64(D_m * Float64(M / d_m)) / Float64(d_m * 4.0))) / l)))))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp = code(w0, M, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h * (((M * D_m) * ((D_m * (M / d_m)) / (d_m * 4.0))) / l))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(D$95$m * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

6. Applied egg-rr 75.6%

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

   1. unpow1 75.6%

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

   2. associate-*l/ 83.8%

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

   3. associate-/l* 83.8%

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

   4. associate-*r/ 83.4%

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

   5. times-frac 83.8%

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

   6. associate-*r/ 83.4%

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

8. Simplified 83.4%

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

   1. div-inv 83.4%

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

   2. div-inv 83.4%

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

   3. metadata-eval 83.4%

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

10. Applied egg-rr 83.4%

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

    1. associate-*l* 83.8%

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

    2. div-inv 83.8%

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

    3. metadata-eval 83.8%

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

    4. div-inv 83.8%

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

    5. unpow2 83.8%

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

    6. frac-times 83.0%

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

    7. associate-*l/ 83.0%

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

    8. frac-times 82.7%

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

12. Applied egg-rr 82.7%

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

    1. associate-/l* 81.1%

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

    2. *-commutative 81.1%

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

    3. associate-*r* 81.1%

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

    4. metadata-eval 81.1%

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

14. Simplified 81.1%

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

15. Final simplification 81.1%

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

Alternative 6: 69.0% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in D around 0 62.2%

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

Reproduce

herbie shell --seed 2024159 
(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))))))