Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 87.0%

Time: 17.3s

Alternatives: 5

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: 80.6% 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: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D_m}{2 \cdot d_m} \leq 10^{-72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left({\left(\sqrt[3]{\left(M \cdot 0.5\right) \cdot \frac{D_m}{d_m}}\right)}^{2} \cdot \sqrt[3]{h}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D_m}{2} \cdot \frac{M}{d_m}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M D_m) (* 2.0 d_m)) 1e-72)
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       (pow (* (pow (cbrt (* (* M 0.5) (/ D_m d_m))) 2.0) (cbrt h)) 3.0)
       l))))
   (* w0 (sqrt (- 1.0 (* (pow (* (/ D_m 2.0) (/ M d_m)) 2.0) (/ h l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M * D_m) / (2.0 * d_m)) <= 1e-72) {
		tmp = w0 * sqrt((1.0 - (pow((pow(cbrt(((M * 0.5) * (D_m / d_m))), 2.0) * cbrt(h)), 3.0) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow(((D_m / 2.0) * (M / d_m)), 2.0) * (h / l))));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M * D_m) / (2.0 * d_m)) <= 1e-72) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((Math.pow(Math.cbrt(((M * 0.5) * (D_m / d_m))), 2.0) * Math.cbrt(h)), 3.0) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((D_m / 2.0) * (M / d_m)), 2.0) * (h / l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M * D_m) / Float64(2.0 * d_m)) <= 1e-72)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64((cbrt(Float64(Float64(M * 0.5) * Float64(D_m / d_m))) ^ 2.0) * cbrt(h)) ^ 3.0) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(D_m / 2.0) * Float64(M / d_m)) ^ 2.0) * Float64(h / l)))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e-72], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[Power[N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $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[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 9.9999999999999997e-73

   1. Initial program 84.6%

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

   3. Simplified 85.0%

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

      1. unpow2 85.0%

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

      2. associate-*l* 86.0%

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

      3. frac-times 85.0%

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

      4. *-commutative 85.0%

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

      5. associate-*l/ 86.0%

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

      6. associate-/r/ 85.5%

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

      7. frac-times 86.0%

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

      8. *-commutative 86.0%

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

      9. associate-*l/ 85.5%

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

      10. associate-/r/ 86.9%

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

      11. associate-*l* 85.5%

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

      12. unpow2 85.5%

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

      13. associate-*r/ 92.7%

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

   6. Applied egg-rr 92.2%

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

      1. add-cube-cbrt 92.2%

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

      2. pow3 92.2%

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

      3. add-sqr-sqrt 92.2%

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

      4. pow2 92.2%

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

      5. unpow2 92.2%

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

      6. sqrt-prod 52.2%

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

      7. add-sqr-sqrt 92.2%

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

      8. div-inv 92.2%

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

      9. metadata-eval 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. cbrt-prod 92.2%

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

      3. associate-/r/ 92.2%

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

      4. pow2 92.2%

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

      5. cbrt-prod 93.8%

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

      6. pow2 93.8%

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

      7. associate-/r/ 93.8%

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

      8. *-commutative 93.8%

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

   10. Applied egg-rr 93.8%

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

   if 9.9999999999999997e-73 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 69.9%

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

   3. Simplified 71.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left({\left(\sqrt[3]{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}\right)}^{2} \cdot \sqrt[3]{h}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D_m}{2 \cdot d_m} \leq 10^{-72}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(h \cdot {\left(M \cdot \frac{D_m}{d_m}\right)}^{2}\right) \cdot -0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D_m}{2} \cdot \frac{M}{d_m}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M D_m) (* 2.0 d_m)) 1e-72)
   (* w0 (sqrt (+ 1.0 (/ (* (* h (pow (* M (/ D_m d_m)) 2.0)) -0.25) l))))
   (* w0 (sqrt (- 1.0 (* (pow (* (/ D_m 2.0) (/ M d_m)) 2.0) (/ h l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M * D_m) / (2.0 * d_m)) <= 1e-72) {
		tmp = w0 * sqrt((1.0 + (((h * pow((M * (D_m / d_m)), 2.0)) * -0.25) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow(((D_m / 2.0) * (M / d_m)), 2.0) * (h / l))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m * d_m) / (2.0d0 * d_m_1)) <= 1d-72) then
        tmp = w0 * sqrt((1.0d0 + (((h * ((m * (d_m / d_m_1)) ** 2.0d0)) * (-0.25d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m / 2.0d0) * (m / d_m_1)) ** 2.0d0) * (h / l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M * D_m) / (2.0 * d_m)) <= 1e-72) {
		tmp = w0 * Math.sqrt((1.0 + (((h * Math.pow((M * (D_m / d_m)), 2.0)) * -0.25) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((D_m / 2.0) * (M / d_m)), 2.0) * (h / l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if ((M * D_m) / (2.0 * d_m)) <= 1e-72:
		tmp = w0 * math.sqrt((1.0 + (((h * math.pow((M * (D_m / d_m)), 2.0)) * -0.25) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((D_m / 2.0) * (M / d_m)), 2.0) * (h / l))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M * D_m) / Float64(2.0 * d_m)) <= 1e-72)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * (Float64(M * Float64(D_m / d_m)) ^ 2.0)) * -0.25) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(D_m / 2.0) * Float64(M / d_m)) ^ 2.0) * Float64(h / l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M * D_m) / (2.0 * d_m)) <= 1e-72)
		tmp = w0 * sqrt((1.0 + (((h * ((M * (D_m / d_m)) ^ 2.0)) * -0.25) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m / 2.0) * (M / d_m)) ^ 2.0) * (h / l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e-72], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[Power[N[(M * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 9.9999999999999997e-73

   1. Initial program 84.6%

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

   3. Simplified 85.0%

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

      1. unpow2 85.0%

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

      2. associate-*l* 86.0%

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

      3. frac-times 85.0%

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

      4. *-commutative 85.0%

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

      5. associate-*l/ 86.0%

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

      6. associate-/r/ 85.5%

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

      7. frac-times 86.0%

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

      8. *-commutative 86.0%

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

      9. associate-*l/ 85.5%

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

      10. associate-/r/ 86.9%

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

      11. associate-*l* 85.5%

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

      12. unpow2 85.5%

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

      13. associate-*r/ 92.7%

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

   6. Applied egg-rr 92.2%

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

      1. add-sqr-sqrt 80.3%

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

      2. add-cube-cbrt 80.3%

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

      3. times-frac 80.3%

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

      4. *-commutative 80.3%

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

      5. sqrt-prod 49.1%

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

      6. unpow2 49.1%

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

      7. sqrt-prod 26.5%

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

      8. add-sqr-sqrt 42.4%

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

      9. div-inv 42.4%

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

      10. metadata-eval 42.4%

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

      11. pow2 42.4%

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

   8. Applied egg-rr 50.3%

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

      1. associate-/l* 50.3%

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

      2. associate-*l/ 50.3%

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

      3. associate-/l* 48.8%

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

      4. associate-/l* 48.8%

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

      5. associate-*l/ 50.2%

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

      6. associate-/l* 50.2%

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

   10. Simplified 50.2%

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

       1. expm1-log1p-u 50.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt{h}}} \cdot \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{\sqrt[3]{\ell}}{\sqrt{h}}}}\right)\right)} \]

       2. expm1-udef 50.1%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt{h}}} \cdot \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{\sqrt[3]{\ell}}{\sqrt{h}}}}\right)} - 1\right)} \]

   12. Applied egg-rr 85.4%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25}{\frac{\ell}{h}}}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 85.4%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25}{\frac{\ell}{h}}}\right)\right)} \]

       2. expm1-log1p 85.5%

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

       3. associate-/r/ 91.7%

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

       4. associate-*l/ 92.2%

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

       5. *-commutative 92.2%

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

       6. sub-neg 92.2%

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

       7. distribute-neg-frac 92.2%

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

       8. associate-*r* 92.2%

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

       9. distribute-rgt-neg-in 92.2%

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

       10. *-commutative 92.2%

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

       11. metadata-eval 92.2%

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

   14. Simplified 92.2%

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

   if 9.9999999999999997e-73 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 69.9%

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

   3. Simplified 71.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-72}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ w0 \cdot \sqrt{1 + \frac{\left(h \cdot {\left(M \cdot \frac{D_m}{d_m}\right)}^{2}\right) \cdot -0.25}{\ell}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 + (((h * math.pow((M * (D_m / d_m)), 2.0)) * -0.25) / l)))

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * (Float64(M * Float64(D_m / d_m)) ^ 2.0)) * -0.25) / l))))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp = code(w0, M, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 + (((h * ((M * (D_m / d_m)) ^ 2.0)) * -0.25) / l)));
end

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Simplified 82.2%

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

   1. unpow2 82.2%

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

   2. associate-*l* 83.3%

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

   3. frac-times 81.4%

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

   4. *-commutative 81.4%

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

   5. associate-*l/ 83.3%

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

   6. associate-/r/ 82.9%

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

   7. frac-times 83.0%

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

   8. *-commutative 83.0%

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

   9. associate-*l/ 82.9%

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

   10. associate-/r/ 84.9%

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

   11. associate-*l* 83.4%

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

   12. unpow2 83.4%

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

   13. associate-*r/ 88.3%

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

6. Applied egg-rr 87.9%

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

   1. add-sqr-sqrt 72.1%

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

   2. add-cube-cbrt 72.1%

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

   3. times-frac 72.1%

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

   4. *-commutative 72.1%

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

   5. sqrt-prod 46.7%

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

   6. unpow2 46.7%

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

   7. sqrt-prod 28.9%

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

   8. add-sqr-sqrt 41.4%

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

   9. div-inv 41.4%

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

   10. metadata-eval 41.4%

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

   11. pow2 41.4%

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

8. Applied egg-rr 47.6%

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

   1. associate-/l* 47.6%

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

   2. associate-*l/ 46.9%

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

   3. associate-/l* 46.5%

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

   4. associate-/l* 46.5%

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

   5. associate-*l/ 46.8%

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

   6. associate-/l* 47.6%

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

10. Simplified 47.6%

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

    1. expm1-log1p-u 47.4%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt{h}}} \cdot \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{\sqrt[3]{\ell}}{\sqrt{h}}}}\right)\right)} \]

    2. expm1-udef 47.4%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt{h}}} \cdot \frac{\frac{D}{\frac{d}{M \cdot 0.5}}}{\frac{\sqrt[3]{\ell}}{\sqrt{h}}}}\right)} - 1\right)} \]

12. Applied egg-rr 83.0%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25}{\frac{\ell}{h}}}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 83.0%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25}{\frac{\ell}{h}}}\right)\right)} \]

    2. expm1-log1p 83.3%

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

    3. associate-/r/ 87.9%

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

    4. associate-*l/ 87.9%

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

    5. *-commutative 87.9%

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

    6. sub-neg 87.9%

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

    7. distribute-neg-frac 87.9%

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

    8. associate-*r* 87.9%

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

    9. distribute-rgt-neg-in 87.9%

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

    10. *-commutative 87.9%

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

    11. metadata-eval 87.9%

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

14. Simplified 87.9%

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

15. Final simplification 87.9%

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

Alternative 4: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.05 \cdot 10^{-17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 2.05e-17) w0 (log1p (expm1 w0))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 2.05e-17) {
		tmp = w0;
	} else {
		tmp = log1p(expm1(w0));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 2.05e-17) {
		tmp = w0;
	} else {
		tmp = Math.log1p(Math.expm1(w0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 2.05e-17:
		tmp = w0
	else:
		tmp = math.log1p(math.expm1(w0))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 2.05e-17)
		tmp = w0;
	else
		tmp = log1p(expm1(w0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 2.05e-17], w0, N[Log[1 + N[(Exp[w0] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.05e-17

   1. Initial program 82.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in D around 0 73.7%

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

   if 2.05e-17 < M

   1. Initial program 75.7%

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

   3. Simplified 84.5%

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

      1. add-sqr-sqrt 43.0%

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

      2. sqrt-unprod 37.3%

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

      3. *-commutative 37.3%

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

      4. *-commutative 37.3%

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

      5. swap-sqr 35.1%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in h around 0 33.5%

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

      1. sqrt-pow1 52.7%

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

      2. metadata-eval 52.7%

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

      3. pow1 52.7%

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

      4. log1p-expm1-u 37.6%

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

   9. Applied egg-rr 37.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.05 \cdot 10^{-17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Simplified 82.2%

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

5. Taylor expanded in D around 0 70.1%

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

6. Final simplification 70.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))