Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.3%

Time: 21.7s

Alternatives: 11

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: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{1}{\frac{D_m \cdot 0.5}{\frac{d}{M}}} \cdot \sqrt{\ell}\\ t_1 := \sqrt[3]{1 - h \cdot \frac{{\left(D_m \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}}\\ \mathbf{if}\;\ell \leq 10^{-306}:\\ \;\;\;\;w0 \cdot \left(t_1 \cdot \sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{t_0} \cdot \frac{h}{t_0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (/ (* D_m 0.5) (/ d M))) (sqrt l)))
        (t_1 (cbrt (- 1.0 (* h (/ (pow (* D_m (/ M (/ d 0.5))) 2.0) l))))))
   (if (<= l 1e-306)
     (* w0 (* t_1 (sqrt t_1)))
     (* w0 (sqrt (- 1.0 (* (/ 1.0 t_0) (/ h t_0))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * sqrt(l);
	double t_1 = cbrt((1.0 - (h * (pow((D_m * (M / (d / 0.5))), 2.0) / l))));
	double tmp;
	if (l <= 1e-306) {
		tmp = w0 * (t_1 * sqrt(t_1));
	} else {
		tmp = w0 * sqrt((1.0 - ((1.0 / t_0) * (h / t_0))));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * Math.sqrt(l);
	double t_1 = Math.cbrt((1.0 - (h * (Math.pow((D_m * (M / (d / 0.5))), 2.0) / l))));
	double tmp;
	if (l <= 1e-306) {
		tmp = w0 * (t_1 * Math.sqrt(t_1));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((1.0 / t_0) * (h / t_0))));
	}
	return tmp;
}

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(Float64(1.0 / Float64(Float64(D_m * 0.5) / Float64(d / M))) * sqrt(l))
	t_1 = cbrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(M / Float64(d / 0.5))) ^ 2.0) / l))))
	tmp = 0.0
	if (l <= 1e-306)
		tmp = Float64(w0 * Float64(t_1 * sqrt(t_1)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(1.0 / t_0) * Float64(h / t_0)))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[l, 1e-306], N[(w0 * N[(t$95$1 * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000003e-306

   1. Initial program 82.2%

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

   3. Simplified 81.5%

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

      1. pow1/2 81.5%

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

      2. *-commutative 81.5%

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

      3. associate-/l* 82.2%

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

      4. *-commutative 82.2%

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

      5. add-cube-cbrt 82.2%

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

      6. unpow-prod-down 82.2%

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

   6. Applied egg-rr 81.4%

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

   8. Simplified 90.3%

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

   if 1.00000000000000003e-306 < l

   1. Initial program 80.0%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in M around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. *-commutative 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*r/ 80.0%

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

      5. associate-*l* 78.4%

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

   7. Simplified 78.4%

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

   9. Applied egg-rr 86.8%

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

       1. div-inv 86.9%

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

       2. associate-*r* 85.2%

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

       3. *-commutative 85.2%

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

       4. *-commutative 85.2%

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

       5. pow-flip 85.2%

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

       6. *-commutative 85.2%

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

       7. *-commutative 85.2%

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

       8. associate-*r* 86.9%

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

       9. metadata-eval 86.9%

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

   11. Applied egg-rr 86.9%

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

       1. *-un-lft-identity 86.9%

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

       2. add-sqr-sqrt 86.9%

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

       3. times-frac 86.9%

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

       4. *-commutative 86.9%

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

       5. sqrt-prod 86.8%

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

       6. sqrt-pow1 78.2%

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

       7. metadata-eval 78.2%

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

       8. unpow-1 78.2%

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

       9. associate-*l/ 78.2%

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

       10. *-commutative 78.2%

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

       11. associate-*r* 78.2%

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

       12. *-commutative 78.2%

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

   13. Applied egg-rr 91.0%

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

       1. associate-/l* 88.4%

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

       2. *-commutative 88.4%

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

       3. associate-/l* 90.0%

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

       4. *-commutative 90.0%

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

   15. Simplified 90.0%

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

4. Final simplification 90.1%

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

Alternative 2: 88.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{1}{\frac{D_m \cdot 0.5}{\frac{d}{M}}} \cdot \sqrt{\ell}\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{0.5}{d} \cdot \left(D_m \cdot M\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{t_0} \cdot \frac{h}{t_0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (/ (* D_m 0.5) (/ d M))) (sqrt l))))
   (if (<= l 4e-307)
     (* w0 (sqrt (- 1.0 (/ h (/ l (pow (* (/ 0.5 d) (* D_m M)) 2.0))))))
     (* w0 (sqrt (- 1.0 (* (/ 1.0 t_0) (/ h t_0))))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * sqrt(l);
	double tmp;
	if (l <= 4e-307) {
		tmp = w0 * sqrt((1.0 - (h / (l / pow(((0.5 / d) * (D_m * M)), 2.0)))));
	} else {
		tmp = w0 * sqrt((1.0 - ((1.0 / t_0) * (h / t_0))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / ((d_m * 0.5d0) / (d / m))) * sqrt(l)
    if (l <= 4d-307) then
        tmp = w0 * sqrt((1.0d0 - (h / (l / (((0.5d0 / d) * (d_m * m)) ** 2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - ((1.0d0 / t_0) * (h / t_0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * Math.sqrt(l);
	double tmp;
	if (l <= 4e-307) {
		tmp = w0 * Math.sqrt((1.0 - (h / (l / Math.pow(((0.5 / d) * (D_m * M)), 2.0)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((1.0 / t_0) * (h / t_0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * math.sqrt(l)
	tmp = 0
	if l <= 4e-307:
		tmp = w0 * math.sqrt((1.0 - (h / (l / math.pow(((0.5 / d) * (D_m * M)), 2.0)))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((1.0 / t_0) * (h / t_0))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(Float64(1.0 / Float64(Float64(D_m * 0.5) / Float64(d / M))) * sqrt(l))
	tmp = 0.0
	if (l <= 4e-307)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / (Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ 2.0))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(1.0 / t_0) * Float64(h / t_0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = (1.0 / ((D_m * 0.5) / (d / M))) * sqrt(l);
	tmp = 0.0;
	if (l <= 4e-307)
		tmp = w0 * sqrt((1.0 - (h / (l / (((0.5 / d) * (D_m * M)) ^ 2.0)))));
	else
		tmp = w0 * sqrt((1.0 - ((1.0 / t_0) * (h / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4e-307], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999964e-307

   1. Initial program 82.2%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in M around 0 82.2%

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

      1. associate-*r/ 82.2%

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

      2. *-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-*r/ 82.2%

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

      5. associate-*l* 81.5%

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

   7. Simplified 81.5%

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

   9. Applied egg-rr 90.3%

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

   if 3.99999999999999964e-307 < l

   1. Initial program 80.0%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in M around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. *-commutative 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*r/ 80.0%

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

      5. associate-*l* 78.4%

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

   7. Simplified 78.4%

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

   9. Applied egg-rr 86.8%

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

       1. div-inv 86.9%

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

       2. associate-*r* 85.2%

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

       3. *-commutative 85.2%

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

       4. *-commutative 85.2%

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

       5. pow-flip 85.2%

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

       6. *-commutative 85.2%

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

       7. *-commutative 85.2%

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

       8. associate-*r* 86.9%

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

       9. metadata-eval 86.9%

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

   11. Applied egg-rr 86.9%

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

       1. *-un-lft-identity 86.9%

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

       2. add-sqr-sqrt 86.9%

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

       3. times-frac 86.9%

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

       4. *-commutative 86.9%

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

       5. sqrt-prod 86.8%

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

       6. sqrt-pow1 78.2%

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

       7. metadata-eval 78.2%

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

       8. unpow-1 78.2%

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

       9. associate-*l/ 78.2%

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

       10. *-commutative 78.2%

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

       11. associate-*r* 78.2%

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

       12. *-commutative 78.2%

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

   13. Applied egg-rr 91.0%

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

       1. associate-/l* 88.4%

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

       2. *-commutative 88.4%

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

       3. associate-/l* 90.0%

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

       4. *-commutative 90.0%

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

   15. Simplified 90.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{1}{\frac{D \cdot 0.5}{\frac{d}{M}}} \cdot \sqrt{\ell}} \cdot \frac{h}{\frac{1}{\frac{D \cdot 0.5}{\frac{d}{M}}} \cdot \sqrt{\ell}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{D_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{\frac{D_m}{2}}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ h l)))))
   (if (<= t_0 2e+247)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (/ (/ D_m 2.0) d)) 2.0)) l)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+247) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * ((D_m / 2.0) / d)), 2.0)) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (h / l))
    if (t_0 <= 2d+247) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * ((d_m / 2.0d0) / d)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+247) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * ((D_m / 2.0) / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = 1.0 - (math.pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 2e+247:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * ((D_m / 2.0) / d)), 2.0)) / l)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2e+247)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(Float64(D_m / 2.0) / d)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = 1.0 - ((((D_m * M) / (d * 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 2e+247)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M * ((D_m / 2.0) / d)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+247], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 1.9999999999999999e247

   1. Initial program 99.9%

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

   if 1.9999999999999999e247 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 40.1%

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

   3. Simplified 41.3%

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

      1. *-commutative 41.3%

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

      2. associate-/l* 40.1%

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

      3. associate-*l/ 67.6%

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

      4. associate-/l* 68.2%

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

      5. div-inv 68.2%

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

      6. clear-num 68.2%

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

      7. associate-/r* 68.2%

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

   6. Applied egg-rr 68.2%

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

4. Final simplification 90.0%

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

Alternative 4: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ h l)) -5e+18)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M (/ (* d 2.0) D_m)) 2.0)))))
   w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l)) <= -5e+18) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M / ((d * 2.0) / D_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (h / l)) <= (-5d+18)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m / ((d * 2.0d0) / d_m)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l)) <= -5e+18) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M / ((d * 2.0) / D_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M) / (d * 2.0)), 2.0) * (h / l)) <= -5e+18:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M / ((d * 2.0) / D_m)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -5e+18)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M / Float64(Float64(d * 2.0) / D_m)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M) / (d * 2.0)) ^ 2.0) * (h / l)) <= -5e+18)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M / ((d * 2.0) / D_m)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+18], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M / N[(N[(d * 2.0), $MachinePrecision] / D$95$m), $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)) < -5e18

   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 69.8%

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

   if -5e18 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 85.6%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in M around 0 95.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.16 \cdot 10^{-153}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{\frac{D_m}{2}}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{{\left(\frac{\sqrt{\ell}}{D_m \cdot \left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= l 1.16e-153)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (/ (/ D_m 2.0) d)) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ h (pow (/ (sqrt l) (* D_m (* M (/ 0.5 d)))) 2.0)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= 1.16e-153) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (h / pow((sqrt(l) / (D_m * (M * (0.5 / d)))), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (l <= 1.16d-153) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * ((d_m / 2.0d0) / d)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (h / ((sqrt(l) / (d_m * (m * (0.5d0 / d)))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (l <= 1.16e-153) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * ((D_m / 2.0) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h / Math.pow((Math.sqrt(l) / (D_m * (M * (0.5 / d)))), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if l <= 1.16e-153:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * ((D_m / 2.0) / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (h / math.pow((math.sqrt(l) / (D_m * (M * (0.5 / d)))), 2.0))))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (l <= 1.16e-153)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(Float64(D_m / 2.0) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / (Float64(sqrt(l) / Float64(D_m * Float64(M * Float64(0.5 / d)))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (l <= 1.16e-153)
		tmp = w0 * sqrt((1.0 - ((h * ((M * ((D_m / 2.0) / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (h / ((sqrt(l) / (D_m * (M * (0.5 / d)))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[l, 1.16e-153], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(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[(h / N[Power[N[(N[Sqrt[l], $MachinePrecision] / N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.16e-153

   1. Initial program 78.6%

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

   3. Simplified 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 78.6%

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

      3. associate-*l/ 88.9%

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

      4. associate-/l* 87.2%

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

      5. div-inv 87.2%

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

      6. clear-num 87.2%

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

      7. associate-/r* 87.2%

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

   6. Applied egg-rr 87.2%

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

   if 1.16e-153 < l

   1. Initial program 85.9%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in M around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r/ 85.9%

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

      5. associate-*l* 84.9%

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

   7. Simplified 84.9%

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

   9. Applied egg-rr 91.5%

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

       1. add-sqr-sqrt 91.5%

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

       2. sqrt-div 91.5%

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

       3. associate-*r* 89.4%

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

       4. *-commutative 89.4%

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

       5. *-commutative 89.4%

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

       6. sqrt-pow1 82.6%

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

       7. metadata-eval 82.6%

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

       8. pow1 82.6%

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

       9. *-commutative 82.6%

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

       10. *-commutative 82.6%

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

       11. associate-*r* 84.8%

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

       12. sqrt-div 84.8%

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

       13. associate-*r* 82.6%

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

       14. *-commutative 82.6%

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

       15. *-commutative 82.6%

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

   11. Applied egg-rr 93.6%

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

       1. unpow2 93.6%

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

       2. associate-*r* 92.5%

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

       3. *-commutative 92.5%

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

   13. Simplified 92.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.16 \cdot 10^{-153}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{{\left(\frac{\sqrt{\ell}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \left(D_m \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-322)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* M (* D_m (/ 0.5 d))) 2.0)))))
   w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-322) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M * (D_m * (0.5 / d))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if ((h / l) <= (-5d-322)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m * (d_m * (0.5d0 / d))) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-322) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M * (D_m * (0.5 / d))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e-322:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M * (D_m * (0.5 / d))), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-322)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M * Float64(D_m * Float64(0.5 / d))) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-322)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * (D_m * (0.5 / d))) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-322], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99006e-322

   1. Initial program 76.4%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in M around 0 76.4%

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

      1. associate-*r/ 76.4%

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

      2. *-commutative 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-*r/ 76.4%

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

      5. associate-*l* 75.9%

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

   7. Simplified 75.9%

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

   if -4.99006e-322 < (/.f64 h l)

   1. Initial program 87.9%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in M around 0 95.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h / (l * pow(((0.5 / d) * (D_m * M)), -2.0)))));
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    code = w0 * sqrt((1.0d0 - (h / (l * (((0.5d0 / d) * (d_m * m)) ** (-2.0d0))))))
end function

Java

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

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - (h / (l * math.pow(((0.5 / d) * (D_m * M)), -2.0)))))

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l * (Float64(Float64(0.5 / d) * Float64(D_m * M)) ^ -2.0))))))
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - (h / (l * (((0.5 / d) * (D_m * M)) ^ -2.0)))));
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 80.1%

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

5. Taylor expanded in M around 0 81.2%

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

   1. associate-*r/ 81.2%

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

   2. *-commutative 81.2%

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

   3. *-commutative 81.2%

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

   4. associate-*r/ 81.2%

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

   5. associate-*l* 80.1%

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

7. Simplified 80.1%

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

9. Applied egg-rr 88.7%

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

    1. div-inv 88.7%

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

    2. associate-*r* 88.0%

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

    3. *-commutative 88.0%

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

    4. *-commutative 88.0%

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

    5. pow-flip 88.0%

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

    6. *-commutative 88.0%

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

    7. *-commutative 88.0%

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

    8. associate-*r* 88.7%

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

    9. metadata-eval 88.7%

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

11. Applied egg-rr 88.7%

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

12. Final simplification 88.7%

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

Alternative 8: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-26}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;{\left({w0}^{6}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= M 3e-26) w0 (pow (pow w0 6.0) 0.16666666666666666)))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 3e-26) {
		tmp = w0;
	} else {
		tmp = pow(pow(w0, 6.0), 0.16666666666666666);
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m <= 3d-26) then
        tmp = w0
    else
        tmp = (w0 ** 6.0d0) ** 0.16666666666666666d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 3e-26) {
		tmp = w0;
	} else {
		tmp = Math.pow(Math.pow(w0, 6.0), 0.16666666666666666);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if M <= 3e-26:
		tmp = w0
	else:
		tmp = math.pow(math.pow(w0, 6.0), 0.16666666666666666)
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (M <= 3e-26)
		tmp = w0;
	else
		tmp = (w0 ^ 6.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (M <= 3e-26)
		tmp = w0;
	else
		tmp = (w0 ^ 6.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[M, 3e-26], w0, N[Power[N[Power[w0, 6.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.00000000000000012e-26

   1. Initial program 83.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in M around 0 74.4%

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

   if 3.00000000000000012e-26 < M

   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 77.0%

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

      1. *-commutative 77.0%

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

      2. associate-/l* 75.6%

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

      3. *-commutative 75.6%

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

      4. add-sqr-sqrt 37.3%

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

      5. sqrt-unprod 29.5%

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

      6. *-commutative 29.5%

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

   6. Applied egg-rr 29.5%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in h around 0 25.8%

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

       1. sqrt-pow1 52.8%

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

       2. metadata-eval 52.8%

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

       3. pow1 52.8%

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

       4. add-exp-log 22.6%

          \[\leadsto \color{blue}{e^{\log w0}} \]

   11. Applied egg-rr 22.6%

       \[\leadsto \color{blue}{e^{\log w0}} \]
   12. Step-by-step derivation

       1. rem-exp-log 52.8%

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

       2. rem-cbrt-cube 37.7%

          \[\leadsto \color{blue}{\sqrt[3]{{w0}^{3}}} \]

       3. unpow1/3 17.4%

          \[\leadsto \color{blue}{{\left({w0}^{3}\right)}^{0.3333333333333333}} \]

       4. sqr-pow 17.4%

          \[\leadsto \color{blue}{{\left({w0}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({w0}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       5. pow-prod-down 16.5%

          \[\leadsto \color{blue}{{\left({w0}^{3} \cdot {w0}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       6. pow-prod-up 16.5%

          \[\leadsto {\color{blue}{\left({w0}^{\left(3 + 3\right)}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       7. metadata-eval 16.5%

          \[\leadsto {\left({w0}^{\color{blue}{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       8. metadata-eval 16.5%

          \[\leadsto {\left({w0}^{6}\right)}^{\color{blue}{0.16666666666666666}} \]

   13. Applied egg-rr 16.5%

       \[\leadsto \color{blue}{{\left({w0}^{6}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-26}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;{\left({w0}^{6}\right)}^{0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{w0}^{2}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= M 1.45e-22) w0 (sqrt (pow w0 2.0))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 1.45e-22) {
		tmp = w0;
	} else {
		tmp = sqrt(pow(w0, 2.0));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m <= 1.45d-22) then
        tmp = w0
    else
        tmp = sqrt((w0 ** 2.0d0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 1.45e-22) {
		tmp = w0;
	} else {
		tmp = Math.sqrt(Math.pow(w0, 2.0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if M <= 1.45e-22:
		tmp = w0
	else:
		tmp = math.sqrt(math.pow(w0, 2.0))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (M <= 1.45e-22)
		tmp = w0;
	else
		tmp = sqrt((w0 ^ 2.0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (M <= 1.45e-22)
		tmp = w0;
	else
		tmp = sqrt((w0 ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[M, 1.45e-22], w0, N[Sqrt[N[Power[w0, 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.4500000000000001e-22

   1. Initial program 83.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in M around 0 74.5%

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

   if 1.4500000000000001e-22 < M

   1. Initial program 75.2%

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

   3. Simplified 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/l* 75.2%

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

      3. *-commutative 75.2%

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

      4. add-sqr-sqrt 37.9%

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

      5. sqrt-unprod 29.8%

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

      6. *-commutative 29.8%

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

   6. Applied egg-rr 29.8%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in h around 0 26.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{w0}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= M 4e+59) w0 (log (exp w0))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 4e+59) {
		tmp = w0;
	} else {
		tmp = log(exp(w0));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m <= 4d+59) then
        tmp = w0
    else
        tmp = log(exp(w0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (M <= 4e+59) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(w0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if M <= 4e+59:
		tmp = w0
	else:
		tmp = math.log(math.exp(w0))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (M <= 4e+59)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (M <= 4e+59)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[M, 4e+59], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.99999999999999989e59

   1. Initial program 82.8%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in M around 0 72.2%

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

   if 3.99999999999999989e59 < M

   1. Initial program 72.5%

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

   3. Simplified 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-/l* 72.5%

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

      3. *-commutative 72.5%

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

      4. add-sqr-sqrt 30.6%

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

      5. sqrt-unprod 24.4%

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

      6. *-commutative 24.4%

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

   6. Applied egg-rr 24.4%

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

   8. Simplified 22.1%

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

   9. Taylor expanded in h around 0 27.8%

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

       1. sqrt-pow1 51.5%

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

       2. metadata-eval 51.5%

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

       3. pow1 51.5%

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

       4. add-log-exp 21.3%

          \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   11. Applied egg-rr 21.3%

       \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 80.1%

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

5. Taylor expanded in M around 0 69.0%

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

6. Final simplification 69.0%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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))))))