Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.9%

Time: 20.4s

Alternatives: 5

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot M\_m}{\frac{d\_m}{D\_m}}\\ \mathbf{if}\;2 \cdot d\_m \leq 5 \cdot 10^{-112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D\_m}{\ell \cdot \frac{d\_m}{M\_m}}\right) \cdot \frac{0.5}{\frac{d\_m}{M\_m \cdot \left(D\_m \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot \left(h \cdot t\_0\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 M_m) (/ d_m D_m))))
   (if (<= (* 2.0 d_m) 5e-112)
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* 0.5 (/ D_m (* l (/ d_m M_m))))
         (/ 0.5 (/ d_m (* M_m (* D_m h))))))))
     (* w0 (sqrt (- 1.0 (/ (* t_0 (* h t_0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (0.5 * M_m) / (d_m / D_m);
	double tmp;
	if ((2.0 * d_m) <= 5e-112) {
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * m_m) / (d_m_1 / d_m)
    if ((2.0d0 * d_m_1) <= 5d-112) then
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (d_m / (l * (d_m_1 / m_m)))) * (0.5d0 / (d_m_1 / (m_m * (d_m * h)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((t_0 * (h * t_0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (0.5 * M_m) / (d_m / D_m);
	double tmp;
	if ((2.0 * d_m) <= 5e-112) {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (0.5 * M_m) / (d_m / D_m)
	tmp = 0
	if (2.0 * d_m) <= 5e-112:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * (h * t_0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(0.5 * M_m) / Float64(d_m / D_m))
	tmp = 0.0
	if (Float64(2.0 * d_m) <= 5e-112)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D_m / Float64(l * Float64(d_m / M_m)))) * Float64(0.5 / Float64(d_m / Float64(M_m * Float64(D_m * h))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(h * t_0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (0.5 * M_m) / (d_m / D_m);
	tmp = 0.0;
	if ((2.0 * d_m) <= 5e-112)
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	else
		tmp = w0 * sqrt((1.0 - ((t_0 * (h * t_0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(0.5 * M$95$m), $MachinePrecision] / N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * d$95$m), $MachinePrecision], 5e-112], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D$95$m / N[(l * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(d$95$m / N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 d) < 5.00000000000000044e-112

   1. Initial program 77.7%

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

   3. Simplified 75.2%

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

      1. *-commutative 75.2%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*l/ 84.3%

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

      5. div-inv 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-*l* 81.9%

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

      8. associate-/r* 81.9%

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

      9. metadata-eval 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/l* 75.1%

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

      3. unpow2 75.1%

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

      4. div-inv 75.1%

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

      5. times-frac 85.6%

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

      6. associate-*r/ 85.6%

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

      7. associate-*r/ 85.6%

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

   8. Applied egg-rr 85.6%

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

   9. Taylor expanded in D around 0 81.3%

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

       1. associate-*r/ 81.3%

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

       2. associate-/l* 81.3%

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

       3. associate-*r* 81.3%

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

       4. *-commutative 81.3%

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

       5. associate-*l* 81.1%

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

   11. Simplified 81.1%

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

   12. Taylor expanded in D around 0 80.4%

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

       1. associate-/r* 82.5%

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

       2. associate-/l* 81.1%

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

       3. associate-/l/ 80.1%

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

   14. Simplified 80.1%

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

   if 5.00000000000000044e-112 < (*.f64 2 d)

   1. Initial program 88.6%

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

   3. Simplified 88.6%

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

      1. add-cbrt-cube 87.6%

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

      2. pow1/3 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. unpow1/3 87.6%

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

   8. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. clear-num 87.6%

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

      3. div-inv 87.6%

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

      4. unpow2 87.6%

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

      5. associate-/l* 88.6%

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

      6. associate-*r/ 88.6%

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

      7. associate-*r/ 88.6%

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

   10. Applied egg-rr 88.6%

       \[\leadsto w0 \cdot \sqrt[3]{{\left(1 - \color{blue}{\frac{D \cdot \frac{M \cdot 0.5}{d}}{\frac{\frac{\ell}{h}}{D \cdot \frac{M \cdot 0.5}{d}}}}\right)}^{1.5}} \]
   11. Step-by-step derivation

       1. pow1/3 88.3%

          \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\frac{\frac{\ell}{h}}{D \cdot \frac{M \cdot 0.5}{d}}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 89.6%

          \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\frac{\frac{\ell}{h}}{D \cdot \frac{M \cdot 0.5}{d}}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 89.6%

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

       4. pow1/2 89.6%

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

       5. associate-/r/ 89.6%

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

       6. associate-*r/ 88.7%

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

       7. *-commutative 88.7%

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

       8. associate-/l* 88.2%

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

       9. associate-*r/ 87.3%

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

       10. *-commutative 87.3%

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

       11. associate-/l* 88.2%

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

   12. Applied egg-rr 88.2%

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

       1. associate-/r/ 94.0%

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

       2. associate-/r/ 94.0%

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

       3. *-commutative 94.0%

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

       4. associate-/r/ 95.3%

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

       5. *-commutative 95.3%

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

       6. associate-*r* 95.3%

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

       7. associate-*l/ 95.4%

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

       8. *-commutative 95.4%

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

       9. associate-/r/ 94.1%

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

       10. *-commutative 94.1%

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

       11. *-commutative 94.1%

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

       12. *-commutative 94.1%

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

       13. associate-/r/ 94.1%

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

       14. *-commutative 94.1%

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

   14. Applied egg-rr 94.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{-112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D}{\ell \cdot \frac{d}{M}}\right) \cdot \frac{0.5}{\frac{d}{M \cdot \left(D \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.5 \cdot M}{\frac{d}{D}} \cdot \left(h \cdot \frac{0.5 \cdot M}{\frac{d}{D}}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 10^{+167}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D\_m}{\ell \cdot \frac{d\_m}{M\_m}}\right) \cdot \frac{0.5}{\frac{d\_m}{M\_m \cdot \left(D\_m \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 1e+167)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* 0.5 (/ D_m (* l (/ d_m M_m))))
       (/ 0.5 (/ d_m (* M_m (* D_m h))))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1e+167) {
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 1d+167) then
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (d_m / (l * (d_m_1 / m_m)))) * (0.5d0 / (d_m_1 / (m_m * (d_m * h)))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1e+167) {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 1e+167:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 1e+167)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D_m / Float64(l * Float64(d_m / M_m)))) * Float64(0.5 / Float64(d_m / Float64(M_m * Float64(D_m * h))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 1e+167)
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 1e+167], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D$95$m / N[(l * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(d$95$m / N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1e167

   1. Initial program 80.3%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. frac-times 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*l/ 86.2%

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

      5. div-inv 86.2%

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

      6. *-commutative 86.2%

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

      7. associate-*l* 84.4%

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

      8. associate-/r* 84.4%

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

      9. metadata-eval 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-/l* 78.5%

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

      3. unpow2 78.5%

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

      4. div-inv 78.5%

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

      5. times-frac 87.1%

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

      6. associate-*r/ 87.1%

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

      7. associate-*r/ 87.1%

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

   8. Applied egg-rr 87.1%

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

   9. Taylor expanded in D around 0 83.5%

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

       1. associate-*r/ 83.5%

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

       2. associate-/l* 83.5%

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

       3. associate-*r* 83.5%

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

       4. *-commutative 83.5%

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

       5. associate-*l* 83.3%

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

   11. Simplified 83.3%

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

   12. Taylor expanded in D around 0 81.8%

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

       1. associate-/r* 84.4%

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

       2. associate-/l* 83.3%

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

       3. associate-/l/ 82.6%

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

   14. Simplified 82.6%

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

   if 1e167 < d

   1. Initial program 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in D around 0 91.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{+167}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D}{\ell \cdot \frac{d}{M}}\right) \cdot \frac{0.5}{\frac{d}{M \cdot \left(D \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{0.5 \cdot M\_m}{d\_m}\\ \mathbf{if}\;D\_m \leq 5 \cdot 10^{+29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D\_m}{\ell \cdot \frac{d\_m}{M\_m}}\right) \cdot \frac{0.5}{\frac{d\_m}{M\_m \cdot \left(D\_m \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ (* 0.5 M_m) d_m))))
   (if (<= D_m 5e+29)
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* 0.5 (/ D_m (* l (/ d_m M_m))))
         (/ 0.5 (/ d_m (* M_m (* D_m h))))))))
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * ((0.5 * M_m) / d_m);
	double tmp;
	if (D_m <= 5e+29) {
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * ((0.5d0 * m_m) / d_m_1)
    if (d_m <= 5d+29) then
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (d_m / (l * (d_m_1 / m_m)))) * (0.5d0 / (d_m_1 / (m_m * (d_m * h)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * ((0.5 * M_m) / d_m);
	double tmp;
	if (D_m <= 5e+29) {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m * ((0.5 * M_m) / d_m)
	tmp = 0
	if D_m <= 5e+29:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(Float64(0.5 * M_m) / d_m))
	tmp = 0.0
	if (D_m <= 5e+29)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D_m / Float64(l * Float64(d_m / M_m)))) * Float64(0.5 / Float64(d_m / Float64(M_m * Float64(D_m * h))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m * ((0.5 * M_m) / d_m);
	tmp = 0.0;
	if (D_m <= 5e+29)
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m / (l * (d_m / M_m)))) * (0.5 / (d_m / (M_m * (D_m * h)))))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(N[(0.5 * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 5e+29], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D$95$m / N[(l * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(d$95$m / N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 5.0000000000000001e29

   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 80.9%

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

      1. *-commutative 80.9%

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

      2. frac-times 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-*l/ 91.1%

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

      5. div-inv 91.1%

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

      6. *-commutative 91.1%

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

      7. associate-*l* 89.3%

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

      8. associate-/r* 89.3%

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

      9. metadata-eval 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-/l* 81.4%

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

      3. unpow2 81.4%

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

      4. div-inv 81.4%

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

      5. times-frac 91.1%

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

      6. associate-*r/ 91.1%

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

      7. associate-*r/ 91.1%

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

   8. Applied egg-rr 91.1%

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

   9. Taylor expanded in D around 0 86.5%

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

       1. associate-*r/ 86.5%

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

       2. associate-/l* 86.5%

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

       3. associate-*r* 87.9%

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

       4. *-commutative 87.9%

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

       5. associate-*l* 86.3%

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

   11. Simplified 86.3%

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

   12. Taylor expanded in D around 0 84.8%

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

       1. associate-/r* 87.8%

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

       2. associate-/l* 86.3%

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

       3. associate-/l/ 84.7%

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

   14. Simplified 84.7%

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

   if 5.0000000000000001e29 < D

   1. Initial program 79.0%

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

   3. Simplified 79.0%

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

      1. *-commutative 79.0%

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

      2. frac-times 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*l/ 79.4%

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

      5. div-inv 79.4%

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

      6. *-commutative 79.4%

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

      7. associate-*l* 79.4%

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

      8. associate-/r* 79.4%

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

      9. metadata-eval 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*l/ 79.0%

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

      2. unpow2 79.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r/ 83.0%

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

      5. associate-*r/ 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{+29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{D}{\ell \cdot \frac{d}{M}}\right) \cdot \frac{0.5}{\frac{d}{M \cdot \left(D \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{0.5 \cdot M}{d}\right) \cdot \left(\left(D \cdot \frac{0.5 \cdot M}{d}\right) \cdot \frac{h}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{0.5 \cdot M\_m}{d\_m}\\ w0 \cdot \sqrt{1 - \frac{t\_0}{\ell} \cdot \left(h \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

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

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * ((0.5 * M_m) / d_m);
	return w0 * sqrt((1.0 - ((t_0 / l) * (h * t_0))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = d_m * ((0.5d0 * m_m) / d_m_1)
    code = w0 * sqrt((1.0d0 - ((t_0 / l) * (h * t_0))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * ((0.5 * M_m) / d_m);
	return w0 * Math.sqrt((1.0 - ((t_0 / l) * (h * t_0))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m * ((0.5 * M_m) / d_m)
	return w0 * math.sqrt((1.0 - ((t_0 / l) * (h * t_0))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(Float64(0.5 * M_m) / d_m))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(h * t_0)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m * ((0.5 * M_m) / d_m);
	tmp = w0 * sqrt((1.0 - ((t_0 / l) * (h * t_0))));
end

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 80.6%

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

   1. *-commutative 80.6%

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

   2. frac-times 82.1%

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

   3. *-commutative 82.1%

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

   4. associate-*l/ 88.8%

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

   5. div-inv 88.8%

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

   6. *-commutative 88.8%

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

   7. associate-*l* 87.3%

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

   8. associate-/r* 87.3%

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

   9. metadata-eval 87.3%

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

6. Applied egg-rr 87.3%

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

   1. *-commutative 87.3%

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

   2. associate-/l* 80.5%

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

   3. unpow2 80.5%

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

   4. div-inv 80.5%

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

   5. times-frac 89.5%

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

   6. associate-*r/ 89.5%

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

   7. associate-*r/ 89.5%

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

8. Applied egg-rr 89.5%

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

   1. div-inv 89.5%

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

   2. associate-*r/ 88.4%

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

   3. *-commutative 88.4%

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

   4. associate-/l* 87.2%

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

   5. inv-pow 87.2%

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

   6. pow-flip 87.2%

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

   7. metadata-eval 87.2%

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

   8. pow1 87.2%

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

10. Applied egg-rr 87.2%

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

    1. *-commutative 87.2%

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

    2. associate-/r/ 89.5%

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

    3. *-commutative 89.5%

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

12. Simplified 89.5%

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

13. Final simplification 89.5%

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

Alternative 5: 67.0% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 80.6%

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

5. Taylor expanded in D around 0 65.6%

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

6. Final simplification 65.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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))))))