Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 89.9%

Time: 18.7s

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: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot \left(M\_m \cdot D\_m\right)\\ \mathbf{if}\;M\_m \leq 5 \cdot 10^{-58}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot t\_0\right) \cdot \frac{t\_0}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)}{\ell}\right) \cdot \left(M\_m \cdot \frac{0.5}{\frac{d}{D\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) (* M_m D_m))))
   (if (<= M_m 5e-58)
     (* w0 (sqrt (- 1.0 (* (* h t_0) (/ t_0 l)))))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* h (/ (* M_m (* 0.5 (/ D_m d))) l))
         (* M_m (/ 0.5 (/ d D_m))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * (M_m * D_m);
	double tmp;
	if (M_m <= 5e-58) {
		tmp = w0 * sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * (0.5 * (D_m / d))) / l)) * (M_m * (0.5 / (d / D_m))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / d) * (m_m * d_m)
    if (m_m <= 5d-58) then
        tmp = w0 * sqrt((1.0d0 - ((h * t_0) * (t_0 / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * (0.5d0 * (d_m / d))) / l)) * (m_m * (0.5d0 / (d / d_m))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * (M_m * D_m);
	double tmp;
	if (M_m <= 5e-58) {
		tmp = w0 * Math.sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * ((M_m * (0.5 * (D_m / d))) / l)) * (M_m * (0.5 / (d / D_m))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (0.5 / d) * (M_m * D_m)
	tmp = 0
	if M_m <= 5e-58:
		tmp = w0 * math.sqrt((1.0 - ((h * t_0) * (t_0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * ((M_m * (0.5 * (D_m / d))) / l)) * (M_m * (0.5 / (d / D_m))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * Float64(M_m * D_m))
	tmp = 0.0
	if (M_m <= 5e-58)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * t_0) * Float64(t_0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d))) / l)) * Float64(M_m * Float64(0.5 / Float64(d / D_m)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (0.5 / d) * (M_m * D_m);
	tmp = 0.0;
	if (M_m <= 5e-58)
		tmp = w0 * sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * (0.5 * (D_m / d))) / l)) * (M_m * (0.5 / (d / D_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 5e-58], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999977e-58

   1. Initial program 79.9%

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

   3. Simplified 80.5%

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

      1. associate-*r/ 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-*l/ 86.1%

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

      4. associate-*r/ 86.4%

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

      5. div-inv 86.4%

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

      6. metadata-eval 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. associate-/l* 79.9%

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

      2. *-commutative 79.9%

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

      3. unpow2 79.9%

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

      4. associate-*r* 81.4%

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

      5. associate-*l* 81.4%

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

      6. clear-num 81.4%

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

      7. un-div-inv 81.4%

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

      8. associate-*l* 81.4%

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

      9. clear-num 81.4%

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

      10. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

      1. associate-*r/ 81.4%

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

      2. associate-*l/ 89.4%

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

      3. div-inv 89.4%

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

      4. clear-num 89.4%

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

      5. associate-*l* 89.4%

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

   10. Applied egg-rr 89.4%

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

       1. *-un-lft-identity 89.4%

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

       2. associate-*l/ 88.5%

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

       3. clear-num 88.5%

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

       4. div-inv 88.5%

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

       5. associate-*r* 86.1%

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

       6. associate-/r/ 86.1%

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

       7. associate-/r/ 86.1%

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

   12. Applied egg-rr 86.1%

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

       1. *-lft-identity 86.1%

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

       2. associate-/l* 86.6%

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

       3. associate-*l* 88.9%

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

       4. *-commutative 88.9%

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

       5. associate-*l* 87.0%

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

       6. *-commutative 87.0%

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

       7. associate-*l* 89.0%

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

   14. Simplified 89.0%

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

   if 4.99999999999999977e-58 < M

   1. Initial program 78.2%

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

   3. Simplified 76.8%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*l/ 80.2%

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

      4. associate-*r/ 79.4%

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

      5. div-inv 79.4%

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

      6. metadata-eval 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-/l* 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. associate-*r* 82.1%

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

      5. associate-*l* 82.1%

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

      6. clear-num 81.6%

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

      7. un-div-inv 81.6%

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

      8. associate-*l* 81.6%

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

      9. clear-num 81.6%

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

      10. un-div-inv 81.6%

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

   8. Applied egg-rr 81.6%

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

      1. associate-*r/ 81.6%

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

      2. associate-*l/ 90.2%

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

      3. div-inv 90.2%

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

      4. clear-num 90.2%

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

      5. associate-*l* 90.2%

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

   10. Applied egg-rr 90.2%

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

       1. associate-/l* 90.2%

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

   12. Simplified 90.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-58}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{\frac{0.5}{d} \cdot \left(M \cdot D\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\ell}\right) \cdot \left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(D\_m \cdot \frac{M\_m \cdot h}{d \cdot \ell}\right)\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{0.5}{\frac{d}{D\_m}}\right) \cdot \left(M\_m \cdot \left(\left(0.5 \cdot \frac{D\_m}{d}\right) \cdot \frac{h}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -1e+135)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* 0.5 (* D_m (/ (* M_m h) (* d l)))) (/ (* D_m (* M_m 0.5)) d)))))
   (if (<= (/ h l) -2e-296)
     (*
      w0
      (sqrt
       (-
        1.0
        (* (* M_m (/ 0.5 (/ d D_m))) (* M_m (* (* 0.5 (/ D_m d)) (/ h l)))))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+135) {
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m * ((M_m * h) / (d * l)))) * ((D_m * (M_m * 0.5)) / d))));
	} else if ((h / l) <= -2e-296) {
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-1d+135)) then
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (d_m * ((m_m * h) / (d * l)))) * ((d_m * (m_m * 0.5d0)) / d))))
    else if ((h / l) <= (-2d-296)) then
        tmp = w0 * sqrt((1.0d0 - ((m_m * (0.5d0 / (d / d_m))) * (m_m * ((0.5d0 * (d_m / d)) * (h / l))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+135) {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (D_m * ((M_m * h) / (d * l)))) * ((D_m * (M_m * 0.5)) / d))));
	} else if ((h / l) <= -2e-296) {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -1e+135:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (D_m * ((M_m * h) / (d * l)))) * ((D_m * (M_m * 0.5)) / d))))
	elif (h / l) <= -2e-296:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+135)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D_m * Float64(Float64(M_m * h) / Float64(d * l)))) * Float64(Float64(D_m * Float64(M_m * 0.5)) / d)))));
	elseif (Float64(h / l) <= -2e-296)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(0.5 / Float64(d / D_m))) * Float64(M_m * Float64(Float64(0.5 * Float64(D_m / d)) * Float64(h / l)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+135)
		tmp = w0 * sqrt((1.0 - ((0.5 * (D_m * ((M_m * h) / (d * l)))) * ((D_m * (M_m * 0.5)) / d))));
	elseif ((h / l) <= -2e-296)
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+135], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D$95$m * N[(N[(M$95$m * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-296], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -9.99999999999999962e134

   1. Initial program 70.9%

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

   3. Simplified 67.6%

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

      1. associate-*r/ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*l/ 76.7%

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

      4. associate-*r/ 77.0%

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

      5. div-inv 77.0%

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

      6. metadata-eval 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. associate-/l* 69.9%

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

      2. *-commutative 69.9%

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

      3. unpow2 69.9%

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

      4. associate-*r* 70.1%

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

      5. associate-*l* 70.0%

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

      6. clear-num 69.4%

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

      7. un-div-inv 69.4%

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

      8. associate-*l* 69.4%

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

      9. clear-num 69.4%

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

      10. un-div-inv 69.4%

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

   8. Applied egg-rr 69.4%

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

   9. Taylor expanded in h around 0 69.3%

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

       1. associate-/l* 69.4%

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

   11. Simplified 69.4%

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

       1. div-inv 69.4%

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

       2. clear-num 69.4%

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

       3. associate-*l* 69.4%

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

       4. associate-*r/ 69.4%

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

   13. Applied egg-rr 69.4%

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

   if -9.99999999999999962e134 < (/.f64 h l) < -2e-296

   1. Initial program 86.0%

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

   3. Simplified 86.0%

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

      1. associate-*r/ 85.0%

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

      2. *-commutative 85.0%

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

      3. associate-*l/ 85.0%

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

      4. associate-*r/ 84.0%

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

      5. div-inv 84.0%

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

      6. metadata-eval 84.0%

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

   6. Applied egg-rr 84.0%

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

      1. associate-/l* 85.0%

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

      2. *-commutative 85.0%

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

      3. unpow2 85.0%

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

      4. associate-*r* 89.0%

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

      5. associate-*l* 89.0%

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

      6. clear-num 89.0%

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

      7. un-div-inv 89.0%

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

      8. associate-*l* 89.0%

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

      9. clear-num 89.0%

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

      10. un-div-inv 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. associate-*r/ 89.0%

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

      2. associate-*l/ 89.0%

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

      3. div-inv 89.0%

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

      4. clear-num 89.0%

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

      5. associate-*l* 89.0%

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

   10. Applied egg-rr 89.0%

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

       1. *-commutative 89.0%

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

       2. associate-*r/ 89.0%

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

       3. associate-*l* 87.1%

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

   12. Simplified 87.1%

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

   if -2e-296 < (/.f64 h l)

   1. Initial program 78.2%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in D around 0 89.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(D \cdot \frac{M \cdot h}{d \cdot \ell}\right)\right) \cdot \frac{D \cdot \left(M \cdot 0.5\right)}{d}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(M \cdot \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot \left(M\_m \cdot D\_m\right)\\ \mathbf{if}\;M\_m \leq 1.25 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot t\_0\right) \cdot \frac{t\_0}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{0.5}{\frac{d}{D\_m}}\right) \cdot \left(M\_m \cdot \left(\left(0.5 \cdot \frac{D\_m}{d}\right) \cdot \frac{h}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) (* M_m D_m))))
   (if (<= M_m 1.25e+70)
     (* w0 (sqrt (- 1.0 (* (* h t_0) (/ t_0 l)))))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* M_m (/ 0.5 (/ d D_m)))
         (* M_m (* (* 0.5 (/ D_m d)) (/ h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * (M_m * D_m);
	double tmp;
	if (M_m <= 1.25e+70) {
		tmp = w0 * sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / d) * (m_m * d_m)
    if (m_m <= 1.25d+70) then
        tmp = w0 * sqrt((1.0d0 - ((h * t_0) * (t_0 / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * (0.5d0 / (d / d_m))) * (m_m * ((0.5d0 * (d_m / d)) * (h / l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * (M_m * D_m);
	double tmp;
	if (M_m <= 1.25e+70) {
		tmp = w0 * Math.sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (0.5 / d) * (M_m * D_m)
	tmp = 0
	if M_m <= 1.25e+70:
		tmp = w0 * math.sqrt((1.0 - ((h * t_0) * (t_0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * Float64(M_m * D_m))
	tmp = 0.0
	if (M_m <= 1.25e+70)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * t_0) * Float64(t_0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(0.5 / Float64(d / D_m))) * Float64(M_m * Float64(Float64(0.5 * Float64(D_m / d)) * Float64(h / l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (0.5 / d) * (M_m * D_m);
	tmp = 0.0;
	if (M_m <= 1.25e+70)
		tmp = w0 * sqrt((1.0 - ((h * t_0) * (t_0 / l))));
	else
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 / (d / D_m))) * (M_m * ((0.5 * (D_m / d)) * (h / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.25e+70], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.2500000000000001e70

   1. Initial program 81.2%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*l/ 87.1%

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

      4. associate-*r/ 87.4%

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

      5. div-inv 87.4%

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

      6. metadata-eval 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. associate-/l* 81.2%

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

      2. *-commutative 81.2%

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

      3. unpow2 81.2%

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

      4. associate-*r* 82.5%

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

      5. associate-*l* 82.5%

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

      6. clear-num 82.5%

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

      7. un-div-inv 82.5%

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

      8. associate-*l* 82.5%

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

      9. clear-num 82.5%

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

      10. un-div-inv 82.5%

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

   8. Applied egg-rr 82.5%

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

      1. associate-*r/ 82.5%

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

      2. associate-*l/ 90.1%

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

      3. div-inv 90.1%

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

      4. clear-num 90.1%

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

      5. associate-*l* 90.1%

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

   10. Applied egg-rr 90.1%

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

       1. *-un-lft-identity 90.1%

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

       2. associate-*l/ 89.3%

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

       3. clear-num 89.3%

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

       4. div-inv 89.3%

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

       5. associate-*r* 87.2%

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

       6. associate-/r/ 87.2%

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

       7. associate-/r/ 87.2%

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

   12. Applied egg-rr 87.2%

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

       1. *-lft-identity 87.2%

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

       2. associate-/l* 87.6%

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

       3. associate-*l* 89.7%

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

       4. *-commutative 89.7%

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

       5. associate-*l* 88.0%

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

       6. *-commutative 88.0%

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

       7. associate-*l* 89.8%

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

   14. Simplified 89.8%

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

   if 1.2500000000000001e70 < M

   1. Initial program 71.2%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 72.0%

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

      2. *-commutative 72.0%

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

      3. associate-*l/ 72.0%

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

      4. associate-*r/ 70.8%

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

      5. div-inv 70.8%

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

      6. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. associate-/l* 70.0%

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

      2. *-commutative 70.0%

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

      3. unpow2 70.0%

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

      4. associate-*r* 77.2%

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

      5. associate-*l* 77.1%

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

      6. clear-num 76.3%

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

      7. un-div-inv 76.3%

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

      8. associate-*l* 76.3%

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

      9. clear-num 76.3%

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

      10. un-div-inv 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. associate-*r/ 76.3%

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

      2. associate-*l/ 87.2%

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

      3. div-inv 87.2%

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

      4. clear-num 87.2%

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

      5. associate-*l* 87.2%

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

   10. Applied egg-rr 87.2%

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

       1. *-commutative 87.2%

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

       2. associate-*r/ 76.3%

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

       3. associate-*l* 74.2%

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

   12. Simplified 74.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{\frac{0.5}{d} \cdot \left(M \cdot D\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(M \cdot \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.5%

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

   1. associate-*r/ 85.2%

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

   2. *-commutative 85.2%

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

   3. associate-*l/ 84.5%

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

   4. associate-*r/ 84.6%

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

   5. div-inv 84.6%

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

   6. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. associate-/l* 79.3%

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

   2. *-commutative 79.3%

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

   3. unpow2 79.3%

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

   4. associate-*r* 81.6%

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

   5. associate-*l* 81.6%

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

   6. clear-num 81.4%

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

   7. un-div-inv 81.4%

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

   8. associate-*l* 81.4%

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

   9. clear-num 81.4%

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

   10. un-div-inv 81.4%

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

8. Applied egg-rr 81.4%

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

9. Taylor expanded in h around 0 80.1%

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

    1. associate-/l* 81.0%

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

11. Simplified 81.0%

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

12. Final simplification 81.0%

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

Alternative 5: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in D around 0 65.4%

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

6. Final simplification 65.4%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))))))