Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 89.1%

Time: 15.4s

Alternatives: 9

Speedup: 1.0×

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: 81.6% 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: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{M_m}{d} \cdot \left(D_m \cdot 0.5\right)\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left({\left(\frac{1}{\sqrt[3]{\frac{2 \cdot \frac{d}{D_m}}{M_m}}}\right)}^{2} \cdot \sqrt[3]{h}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\ell}}\\ \end{array} \end{array} \]

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 (* (* (/ M_m d) (* D_m 0.5)) (sqrt h))))
   (if (<= h -4e-306)
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (pow
          (* (pow (/ 1.0 (cbrt (/ (* 2.0 (/ d D_m)) M_m))) 2.0) (cbrt h))
          3.0)
         l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_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 = ((M_m / d) * (D_m * 0.5)) * sqrt(h);
	double tmp;
	if (h <= -4e-306) {
		tmp = w0 * sqrt((1.0 - (pow((pow((1.0 / cbrt(((2.0 * (d / D_m)) / M_m))), 2.0) * cbrt(h)), 3.0) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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 t_0 = ((M_m / d) * (D_m * 0.5)) * Math.sqrt(h);
	double tmp;
	if (h <= -4e-306) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((Math.pow((1.0 / Math.cbrt(((2.0 * (d / D_m)) / M_m))), 2.0) * Math.cbrt(h)), 3.0) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_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(Float64(Float64(M_m / d) * Float64(D_m * 0.5)) * sqrt(h))
	tmp = 0.0
	if (h <= -4e-306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64((Float64(1.0 / cbrt(Float64(Float64(2.0 * Float64(d / D_m)) / M_m))) ^ 2.0) * cbrt(h)) ^ 3.0) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_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_] := Block[{t$95$0 = N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e-306], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[Power[N[(1.0 / N[Power[N[(N[(2.0 * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[h, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.00000000000000011e-306

   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 84.3%

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

      1. associate-*r/ 89.1%

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

      2. associate-*l/ 89.1%

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

      3. div-inv 89.1%

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

      4. associate-*l* 88.3%

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

      5. associate-/r* 88.3%

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

      6. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. add-cube-cbrt 88.3%

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

      2. pow3 88.3%

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

      3. associate-*r/ 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. cbrt-prod 88.3%

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

   10. Applied egg-rr 89.8%

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

       1. *-commutative 89.8%

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

       2. associate-*r/ 89.8%

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

       3. *-commutative 89.8%

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

       4. associate-*r/ 90.6%

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

       5. associate-/l* 89.8%

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

   12. Simplified 89.8%

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

       1. clear-num 89.8%

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

       2. cbrt-div 89.9%

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

       3. metadata-eval 89.9%

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

       4. *-un-lft-identity 89.9%

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

       5. *-commutative 89.9%

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

       6. times-frac 89.9%

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

       7. metadata-eval 89.9%

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

   14. Applied egg-rr 89.9%

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

   if -4.00000000000000011e-306 < h

   1. Initial program 82.6%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 86.5%

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

      2. associate-*l/ 85.7%

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

      3. div-inv 85.7%

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

      4. associate-*l* 85.7%

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

      5. associate-/r* 85.7%

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

      6. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. add-cube-cbrt 85.7%

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

      2. pow3 85.7%

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

      3. associate-*r/ 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. cbrt-prod 85.6%

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

   10. Applied egg-rr 87.6%

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

       1. *-commutative 87.6%

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

       2. associate-*r/ 87.6%

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

       3. *-commutative 87.6%

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

       4. associate-*r/ 88.4%

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

       5. associate-/l* 87.7%

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

   12. Simplified 87.7%

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

       1. add-sqr-sqrt 87.7%

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

       2. *-un-lft-identity 87.7%

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

       3. times-frac 87.7%

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

   14. Applied egg-rr 90.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left({\left(\frac{1}{\sqrt[3]{\frac{2 \cdot \frac{d}{D}}{M}}}\right)}^{2} \cdot \sqrt[3]{h}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{M_m}{d} \cdot \left(D_m \cdot 0.5\right)\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{h} \cdot {\left(\sqrt[3]{\frac{M_m}{\frac{d}{D_m \cdot 0.5}}}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\ell}}\\ \end{array} \end{array} \]

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 (* (* (/ M_m d) (* D_m 0.5)) (sqrt h))))
   (if (<= h -4e-306)
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (pow (* (cbrt h) (pow (cbrt (/ M_m (/ d (* D_m 0.5)))) 2.0)) 3.0)
         l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_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 = ((M_m / d) * (D_m * 0.5)) * sqrt(h);
	double tmp;
	if (h <= -4e-306) {
		tmp = w0 * sqrt((1.0 - (pow((cbrt(h) * pow(cbrt((M_m / (d / (D_m * 0.5)))), 2.0)), 3.0) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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 t_0 = ((M_m / d) * (D_m * 0.5)) * Math.sqrt(h);
	double tmp;
	if (h <= -4e-306) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((Math.cbrt(h) * Math.pow(Math.cbrt((M_m / (d / (D_m * 0.5)))), 2.0)), 3.0) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_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(Float64(Float64(M_m / d) * Float64(D_m * 0.5)) * sqrt(h))
	tmp = 0.0
	if (h <= -4e-306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(cbrt(h) * (cbrt(Float64(M_m / Float64(d / Float64(D_m * 0.5)))) ^ 2.0)) ^ 3.0) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_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_] := Block[{t$95$0 = N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e-306], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[Power[h, 1/3], $MachinePrecision] * N[Power[N[Power[N[(M$95$m / N[(d / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.00000000000000011e-306

   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 84.3%

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

      1. associate-*r/ 89.1%

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

      2. associate-*l/ 89.1%

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

      3. div-inv 89.1%

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

      4. associate-*l* 88.3%

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

      5. associate-/r* 88.3%

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

      6. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. add-cube-cbrt 88.3%

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

      2. pow3 88.3%

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

      3. associate-*r/ 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. cbrt-prod 88.3%

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

   10. Applied egg-rr 89.8%

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

       1. *-commutative 89.8%

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

       2. associate-*r/ 89.8%

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

       3. *-commutative 89.8%

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

       4. associate-*r/ 90.6%

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

       5. associate-/l* 89.8%

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

   12. Simplified 89.8%

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

   if -4.00000000000000011e-306 < h

   1. Initial program 82.6%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 86.5%

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

      2. associate-*l/ 85.7%

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

      3. div-inv 85.7%

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

      4. associate-*l* 85.7%

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

      5. associate-/r* 85.7%

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

      6. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. add-cube-cbrt 85.7%

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

      2. pow3 85.7%

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

      3. associate-*r/ 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. cbrt-prod 85.6%

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

   10. Applied egg-rr 87.6%

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

       1. *-commutative 87.6%

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

       2. associate-*r/ 87.6%

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

       3. *-commutative 87.6%

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

       4. associate-*r/ 88.4%

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

       5. associate-/l* 87.7%

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

   12. Simplified 87.7%

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

       1. add-sqr-sqrt 87.7%

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

       2. *-un-lft-identity 87.7%

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

       3. times-frac 87.7%

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

   14. Applied egg-rr 90.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{h} \cdot {\left(\sqrt[3]{\frac{M}{\frac{d}{D \cdot 0.5}}}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.7× speedup

Math

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

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
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) 2e-5)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ M_m (* 2.0 d))) 2.0)))))
   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 tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 2e-5) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * (M_m / (2.0 * d))), 2.0))));
	} else {
		tmp = w0;
	}
	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 * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= 2d-5) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * (m_m / (2.0d0 * d))) ** 2.0d0))))
    else
        tmp = w0
    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 ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 2e-5) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * (M_m / (2.0 * d))), 2.0))));
	} 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):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 2e-5:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D_m * (M_m / (2.0 * d))), 2.0))))
	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)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 2e-5)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / Float64(2.0 * d))) ^ 2.0)))));
	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)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= 2e-5)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * (M_m / (2.0 * d))) ^ 2.0))));
	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_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-5], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2.00000000000000016e-5

   1. Initial program 88.3%

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

   3. Simplified 88.7%

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

   if 2.00000000000000016e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in M around 0 60.2%

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

4. Final simplification 87.1%

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

Alternative 4: 87.6% accurate, 0.7× speedup

Math

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

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 (* (sqrt h) (* M_m (* 0.5 (/ D_m d))))))
   (if (<= h -1e-310)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (/ (/ D_m 2.0) d)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_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 = sqrt(h) * (M_m * (0.5 * (D_m / d)));
	double tmp;
	if (h <= -1e-310) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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 = sqrt(h) * (m_m * (0.5d0 * (d_m / d)))
    if (h <= (-1d-310)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * ((d_m / 2.0d0) / d)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 / 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 = Math.sqrt(h) * (M_m * (0.5 * (D_m / d)));
	double tmp;
	if (h <= -1e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_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 = math.sqrt(h) * (M_m * (0.5 * (D_m / d)))
	tmp = 0
	if h <= -1e-310:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_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(sqrt(h) * Float64(M_m * Float64(0.5 * Float64(D_m / d))))
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m * Float64(Float64(D_m / 2.0) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_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 = sqrt(h) * (M_m * (0.5 * (D_m / d)));
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * ((D_m / 2.0) / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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[(N[Sqrt[h], $MachinePrecision] * N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   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 84.3%

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

      1. associate-*r/ 89.1%

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

      2. associate-*l/ 89.1%

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

      3. div-inv 89.1%

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

      4. associate-*l* 88.3%

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

      5. associate-/r* 88.3%

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

      6. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in M around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-/l* 89.1%

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

      3. metadata-eval 89.1%

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

      4. times-frac 89.1%

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

      5. *-rgt-identity 89.1%

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

      6. associate-/l/ 89.1%

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

      7. associate-/r/ 88.3%

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

   9. Simplified 88.3%

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

   if -9.999999999999969e-311 < h

   1. Initial program 82.6%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 86.5%

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

      2. associate-*l/ 85.7%

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

      3. div-inv 85.7%

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

      4. associate-*l* 85.7%

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

      5. associate-/r* 85.7%

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

      6. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in M around 0 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/l* 86.5%

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

      3. metadata-eval 86.5%

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

      4. times-frac 86.5%

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

      5. *-rgt-identity 86.5%

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

      6. associate-/l/ 86.5%

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

      7. associate-/r/ 85.8%

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

   9. Simplified 85.8%

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

       1. div-inv 85.8%

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

       2. metadata-eval 85.8%

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

       3. *-commutative 85.8%

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

       4. add-sqr-sqrt 85.8%

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

       5. *-un-lft-identity 85.8%

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

       6. times-frac 85.8%

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

   11. Applied egg-rr 89.9%

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

4. Final simplification 89.2%

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

Alternative 5: 88.4% accurate, 0.7× speedup

Math

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

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 (* (* (/ M_m d) (* D_m 0.5)) (sqrt h))))
   (if (<= h -1e-310)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (/ (/ D_m 2.0) d)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_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 = ((M_m / d) * (D_m * 0.5)) * sqrt(h);
	double tmp;
	if (h <= -1e-310) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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 = ((m_m / d) * (d_m * 0.5d0)) * sqrt(h)
    if (h <= (-1d-310)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * ((d_m / 2.0d0) / d)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 / 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 = ((M_m / d) * (D_m * 0.5)) * Math.sqrt(h);
	double tmp;
	if (h <= -1e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_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 = ((M_m / d) * (D_m * 0.5)) * math.sqrt(h)
	tmp = 0
	if h <= -1e-310:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_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(Float64(Float64(M_m / d) * Float64(D_m * 0.5)) * sqrt(h))
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m * Float64(Float64(D_m / 2.0) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_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 = ((M_m / d) * (D_m * 0.5)) * sqrt(h);
	tmp = 0.0;
	if (h <= -1e-310)
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * ((D_m / 2.0) / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_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[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   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 84.3%

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

      1. associate-*r/ 89.1%

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

      2. associate-*l/ 89.1%

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

      3. div-inv 89.1%

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

      4. associate-*l* 88.3%

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

      5. associate-/r* 88.3%

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

      6. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in M around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-/l* 89.1%

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

      3. metadata-eval 89.1%

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

      4. times-frac 89.1%

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

      5. *-rgt-identity 89.1%

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

      6. associate-/l/ 89.1%

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

      7. associate-/r/ 88.3%

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

   9. Simplified 88.3%

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

   if -9.999999999999969e-311 < h

   1. Initial program 82.6%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 86.5%

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

      2. associate-*l/ 85.7%

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

      3. div-inv 85.7%

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

      4. associate-*l* 85.7%

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

      5. associate-/r* 85.7%

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

      6. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. add-cube-cbrt 85.7%

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

      2. pow3 85.7%

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

      3. associate-*r/ 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. cbrt-prod 85.6%

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

   10. Applied egg-rr 87.6%

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

       1. *-commutative 87.6%

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

       2. associate-*r/ 87.6%

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

       3. *-commutative 87.6%

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

       4. associate-*r/ 88.4%

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

       5. associate-/l* 87.7%

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

   12. Simplified 87.7%

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

       1. add-sqr-sqrt 87.7%

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

       2. *-un-lft-identity 87.7%

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

       3. times-frac 87.7%

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

   14. Applied egg-rr 90.7%

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

4. Final simplification 89.6%

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

Alternative 6: 86.1% accurate, 1.0× speedup

Math

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

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 (sqrt (- 1.0 (/ (* h (pow (* M_m (* D_m (/ 0.5 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) {
	return w0 * sqrt((1.0 - ((h * pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
}

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 * sqrt((1.0d0 - ((h * ((m_m * (d_m * (0.5d0 / d))) ** 2.0d0)) / l)))
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 * Math.sqrt((1.0 - ((h * Math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
}

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 * math.sqrt((1.0 - ((h * math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)))

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 Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m * Float64(D_m * Float64(0.5 / d))) ^ 2.0)) / l))))
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 * sqrt((1.0 - ((h * ((M_m * (D_m * (0.5 / d))) ^ 2.0)) / l)));
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Simplified 83.8%

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

   1. associate-*r/ 87.7%

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

   2. associate-*l/ 87.4%

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

   3. div-inv 87.3%

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

   4. associate-*l* 87.0%

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

   5. associate-/r* 87.0%

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

   6. metadata-eval 87.0%

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

6. Applied egg-rr 87.0%

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

7. Final simplification 87.0%

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

Alternative 7: 86.1% accurate, 1.0× speedup

Math

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

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 (sqrt (- 1.0 (/ (* h (pow (* M_m (/ (/ D_m 2.0) 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) {
	return w0 * sqrt((1.0 - ((h * pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
}

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 * sqrt((1.0d0 - ((h * ((m_m * ((d_m / 2.0d0) / d)) ** 2.0d0)) / l)))
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 * Math.sqrt((1.0 - ((h * Math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)));
}

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 * math.sqrt((1.0 - ((h * math.pow((M_m * ((D_m / 2.0) / d)), 2.0)) / l)))

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 Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m * Float64(Float64(D_m / 2.0) / d)) ^ 2.0)) / l))))
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 * sqrt((1.0 - ((h * ((M_m * ((D_m / 2.0) / d)) ^ 2.0)) / l)));
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Simplified 83.8%

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

   1. associate-*r/ 87.7%

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

   2. associate-*l/ 87.4%

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

   3. div-inv 87.3%

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

   4. associate-*l* 87.0%

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

   5. associate-/r* 87.0%

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

   6. metadata-eval 87.0%

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

6. Applied egg-rr 87.0%

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

7. Taylor expanded in M around 0 87.4%

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

   1. *-commutative 87.4%

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

   2. associate-/l* 87.7%

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

   3. metadata-eval 87.7%

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

   4. times-frac 87.7%

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

   5. *-rgt-identity 87.7%

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

   6. associate-/l/ 87.7%

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

   7. associate-/r/ 87.0%

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

9. Simplified 87.0%

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

10. Final simplification 87.0%

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

Alternative 8: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D_m \leq 2.15 \cdot 10^{+198}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \end{array} \]

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 (<= D_m 2.15e+198) w0 (log1p (expm1 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 tmp;
	if (D_m <= 2.15e+198) {
		tmp = w0;
	} else {
		tmp = log1p(expm1(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 tmp;
	if (D_m <= 2.15e+198) {
		tmp = w0;
	} else {
		tmp = Math.log1p(Math.expm1(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):
	tmp = 0
	if D_m <= 2.15e+198:
		tmp = w0
	else:
		tmp = math.log1p(math.expm1(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)
	tmp = 0.0
	if (D_m <= 2.15e+198)
		tmp = w0;
	else
		tmp = log1p(expm1(w0));
	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[D$95$m, 2.15e+198], w0, N[Log[1 + N[(Exp[w0] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.14999999999999991e198

   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 84.7%

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

   5. Taylor expanded in M around 0 71.3%

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

   if 2.14999999999999991e198 < D

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. add-sqr-sqrt 25.6%

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

      2. sqrt-unprod 20.3%

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

      3. *-commutative 20.3%

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

      4. *-commutative 20.3%

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

      5. swap-sqr 19.7%

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

   6. Applied egg-rr 19.7%

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

      1. *-commutative 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*l/ 19.7%

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

      4. associate-*r/ 19.7%

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

   8. Simplified 19.7%

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

   9. Taylor expanded in h around 0 14.1%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 29.2%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 29.2%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 29.2%

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

       4. log1p-expm1-u 28.4%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]

   11. Applied egg-rr 28.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.15 \cdot 10^{+198}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% 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 83.4%

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

3. Simplified 83.8%

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

5. Taylor expanded in M around 0 68.6%

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

6. Final simplification 68.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))