Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.9%

Time: 17.1s

Alternatives: 10

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

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(0.5 \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot \left(D\_m \cdot \left(M\_m \cdot 0.5\right)\right)\right)\right) \cdot h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{\frac{D\_m}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2e-90)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (/ (* (* 0.5 (* (/ (* M_m D_m) d) (* D_m (* M_m 0.5)))) h) l) d))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (/ (/ D_m d) 2.0)) (* h (/ (* 0.5 (* D_m (/ M_m 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) {
	double tmp;
	if (M_m <= 2e-90) {
		tmp = w0 * sqrt((1.0 - ((((0.5 * (((M_m * D_m) / d) * (D_m * (M_m * 0.5)))) * h) / l) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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) :: tmp
    if (m_m <= 2d-90) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * (((m_m * d_m) / d) * (d_m * (m_m * 0.5d0)))) * h) / l) / d)))
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * ((d_m / d) / 2.0d0)) * (h * ((0.5d0 * (d_m * (m_m / d))) / 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 tmp;
	if (M_m <= 2e-90) {
		tmp = w0 * Math.sqrt((1.0 - ((((0.5 * (((M_m * D_m) / d) * (D_m * (M_m * 0.5)))) * h) / l) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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):
	tmp = 0
	if M_m <= 2e-90:
		tmp = w0 * math.sqrt((1.0 - ((((0.5 * (((M_m * D_m) / d) * (D_m * (M_m * 0.5)))) * h) / l) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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)
	tmp = 0.0
	if (M_m <= 2e-90)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(0.5 * Float64(Float64(Float64(M_m * D_m) / d) * Float64(D_m * Float64(M_m * 0.5)))) * h) / l) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(Float64(D_m / d) / 2.0)) * Float64(h * Float64(Float64(0.5 * Float64(D_m * Float64(M_m / d))) / 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)
	tmp = 0.0;
	if (M_m <= 2e-90)
		tmp = w0 * sqrt((1.0 - ((((0.5 * (((M_m * D_m) / d) * (D_m * (M_m * 0.5)))) * h) / l) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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_] := If[LessEqual[M$95$m, 2e-90], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(0.5 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.99999999999999999e-90

   1. Initial program 84.8%

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

   3. Simplified 84.2%

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

      1. unpow2 84.2%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-/r* 84.3%

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

      6. associate-*r/ 83.7%

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

      7. clear-num 83.7%

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

      8. un-div-inv 83.7%

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

      9. associate-/r/ 83.7%

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

      10. div-inv 83.7%

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

      11. *-commutative 83.7%

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

      12. metadata-eval 83.7%

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

      13. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l/ 84.2%

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

   8. Applied egg-rr 84.2%

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

   9. Taylor expanded in D around 0 85.3%

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

       1. associate-*r/ 89.7%

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

       2. associate-*l* 89.7%

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

   11. Applied egg-rr 89.7%

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

   if 1.99999999999999999e-90 < M

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

      1. unpow2 73.8%

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

      2. associate-/r* 73.8%

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

      3. associate-*r/ 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-/r* 73.8%

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

      6. associate-*r/ 71.1%

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

      7. clear-num 71.1%

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

      8. un-div-inv 71.1%

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

      9. associate-/r/ 71.1%

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

      10. div-inv 71.1%

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

      11. *-commutative 71.1%

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

      12. metadata-eval 71.1%

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

      13. associate-*r* 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. associate-*r/ 75.8%

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

      2. associate-/l* 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. pow1 77.2%

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

      2. associate-*l* 78.2%

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

      3. associate-*r* 78.2%

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

   10. Applied egg-rr 78.2%

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

       1. unpow1 78.2%

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

       2. associate-*l/ 78.2%

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

       3. associate-/l* 78.2%

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

       4. *-commutative 78.2%

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

       5. *-commutative 78.2%

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

       6. associate-*r/ 78.2%

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

       7. associate-/l* 79.6%

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

   12. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

       2. associate-/l* 80.9%

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

       3. associate-/r* 80.9%

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

       4. associate-*r/ 80.9%

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

   14. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

       2. associate-/r/ 80.9%

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

       3. associate-/l* 81.0%

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

       4. associate-*r/ 82.3%

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

   16. Simplified 82.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(0.5 \cdot \left(\frac{M \cdot D}{d} \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)\right)\right) \cdot h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot \left(M\_m \cdot 0.5\right)\right) \cdot \frac{D\_m}{2 \cdot \frac{d}{M\_m}}\right) \cdot \frac{h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{\frac{D\_m}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 1.45e-55)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (* (* D_m (* M_m 0.5)) (/ D_m (* 2.0 (/ d M_m)))) (/ h l)) d))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (/ (/ D_m d) 2.0)) (* h (/ (* 0.5 (* D_m (/ M_m 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) {
	double tmp;
	if (d <= 1.45e-55) {
		tmp = w0 * sqrt((1.0 - ((((D_m * (M_m * 0.5)) * (D_m / (2.0 * (d / M_m)))) * (h / l)) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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) :: tmp
    if (d <= 1.45d-55) then
        tmp = w0 * sqrt((1.0d0 - ((((d_m * (m_m * 0.5d0)) * (d_m / (2.0d0 * (d / m_m)))) * (h / l)) / d)))
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * ((d_m / d) / 2.0d0)) * (h * ((0.5d0 * (d_m * (m_m / d))) / 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 tmp;
	if (d <= 1.45e-55) {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m * (M_m * 0.5)) * (D_m / (2.0 * (d / M_m)))) * (h / l)) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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):
	tmp = 0
	if d <= 1.45e-55:
		tmp = w0 * math.sqrt((1.0 - ((((D_m * (M_m * 0.5)) * (D_m / (2.0 * (d / M_m)))) * (h / l)) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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)
	tmp = 0.0
	if (d <= 1.45e-55)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) * Float64(D_m / Float64(2.0 * Float64(d / M_m)))) * Float64(h / l)) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(Float64(D_m / d) / 2.0)) * Float64(h * Float64(Float64(0.5 * Float64(D_m * Float64(M_m / d))) / 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)
	tmp = 0.0;
	if (d <= 1.45e-55)
		tmp = w0 * sqrt((1.0 - ((((D_m * (M_m * 0.5)) * (D_m / (2.0 * (d / M_m)))) * (h / l)) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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_] := If[LessEqual[d, 1.45e-55], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(2.0 * N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.45e-55

   1. Initial program 79.4%

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

   3. Simplified 78.8%

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

      1. unpow2 78.8%

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

      2. associate-/r* 78.8%

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

      3. associate-*r/ 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-/r* 78.8%

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

      6. associate-*r/ 78.8%

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

      7. clear-num 78.8%

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

      8. un-div-inv 78.8%

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

      9. associate-/r/ 78.8%

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

      10. div-inv 78.8%

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

      11. *-commutative 78.8%

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

      12. metadata-eval 78.8%

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

      13. associate-*r* 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. associate-*l/ 78.8%

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

   8. Applied egg-rr 78.8%

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

   if 1.45e-55 < d

   1. Initial program 86.5%

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

   3. Simplified 86.5%

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

      1. unpow2 86.5%

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

      2. associate-/r* 86.5%

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

      3. associate-*r/ 86.5%

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

      4. *-commutative 86.5%

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

      5. associate-/r* 86.5%

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

      6. associate-*r/ 83.2%

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

      7. clear-num 83.2%

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

      8. un-div-inv 83.2%

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

      9. associate-/r/ 83.2%

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

      10. div-inv 83.2%

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

      11. *-commutative 83.2%

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

      12. metadata-eval 83.2%

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

      13. associate-*r* 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 90.1%

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

      2. associate-/l* 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. pow1 92.3%

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

      2. associate-*l* 93.4%

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

      3. associate-*r* 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. unpow1 93.4%

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

       2. associate-*l/ 93.4%

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

       3. associate-/l* 93.4%

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

       4. *-commutative 93.4%

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

       5. *-commutative 93.4%

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

       6. associate-*r/ 93.4%

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

       7. associate-/l* 94.5%

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

   12. Simplified 94.5%

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

       1. *-un-lft-identity 94.5%

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

       2. associate-/l* 95.6%

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

       3. associate-/r* 95.6%

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

       4. associate-*r/ 94.5%

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

   14. Applied egg-rr 94.5%

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

       1. *-lft-identity 94.5%

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

       2. associate-/r/ 94.5%

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

       3. associate-/l* 93.4%

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

       4. associate-*r/ 94.5%

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

   16. Simplified 94.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{D}{2 \cdot \frac{d}{M}}\right) \cdot \frac{h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.1 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell} \cdot \left(\left(D\_m \cdot \left(M\_m \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{\frac{D\_m}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 2.1e-54)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (/ h l) (* (* D_m (* M_m 0.5)) (* 0.5 (/ (* M_m D_m) d)))) d))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (/ (/ D_m d) 2.0)) (* h (/ (* 0.5 (* D_m (/ M_m 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) {
	double tmp;
	if (d <= 2.1e-54) {
		tmp = w0 * sqrt((1.0 - (((h / l) * ((D_m * (M_m * 0.5)) * (0.5 * ((M_m * D_m) / d)))) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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) :: tmp
    if (d <= 2.1d-54) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * ((d_m * (m_m * 0.5d0)) * (0.5d0 * ((m_m * d_m) / d)))) / d)))
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * ((d_m / d) / 2.0d0)) * (h * ((0.5d0 * (d_m * (m_m / d))) / 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 tmp;
	if (d <= 2.1e-54) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * ((D_m * (M_m * 0.5)) * (0.5 * ((M_m * D_m) / d)))) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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):
	tmp = 0
	if d <= 2.1e-54:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * ((D_m * (M_m * 0.5)) * (0.5 * ((M_m * D_m) / d)))) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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)
	tmp = 0.0
	if (d <= 2.1e-54)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(Float64(D_m * Float64(M_m * 0.5)) * Float64(0.5 * Float64(Float64(M_m * D_m) / d)))) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(Float64(D_m / d) / 2.0)) * Float64(h * Float64(Float64(0.5 * Float64(D_m * Float64(M_m / d))) / 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)
	tmp = 0.0;
	if (d <= 2.1e-54)
		tmp = w0 * sqrt((1.0 - (((h / l) * ((D_m * (M_m * 0.5)) * (0.5 * ((M_m * D_m) / d)))) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / d))) / 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_] := If[LessEqual[d, 2.1e-54], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.1e-54

   1. Initial program 79.4%

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

   3. Simplified 78.8%

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

      1. unpow2 78.8%

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

      2. associate-/r* 78.8%

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

      3. associate-*r/ 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-/r* 78.8%

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

      6. associate-*r/ 78.8%

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

      7. clear-num 78.8%

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

      8. un-div-inv 78.8%

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

      9. associate-/r/ 78.8%

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

      10. div-inv 78.8%

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

      11. *-commutative 78.8%

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

      12. metadata-eval 78.8%

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

      13. associate-*r* 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. associate-*l/ 78.8%

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

   8. Applied egg-rr 78.8%

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

   9. Taylor expanded in D around 0 80.5%

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

   if 2.1e-54 < d

   1. Initial program 86.5%

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

   3. Simplified 86.5%

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

      1. unpow2 86.5%

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

      2. associate-/r* 86.5%

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

      3. associate-*r/ 86.5%

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

      4. *-commutative 86.5%

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

      5. associate-/r* 86.5%

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

      6. associate-*r/ 83.2%

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

      7. clear-num 83.2%

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

      8. un-div-inv 83.2%

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

      9. associate-/r/ 83.2%

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

      10. div-inv 83.2%

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

      11. *-commutative 83.2%

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

      12. metadata-eval 83.2%

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

      13. associate-*r* 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 90.1%

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

      2. associate-/l* 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. pow1 92.3%

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

      2. associate-*l* 93.4%

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

      3. associate-*r* 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. unpow1 93.4%

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

       2. associate-*l/ 93.4%

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

       3. associate-/l* 93.4%

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

       4. *-commutative 93.4%

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

       5. *-commutative 93.4%

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

       6. associate-*r/ 93.4%

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

       7. associate-/l* 94.5%

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

   12. Simplified 94.5%

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

       1. *-un-lft-identity 94.5%

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

       2. associate-/l* 95.6%

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

       3. associate-/r* 95.6%

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

       4. associate-*r/ 94.5%

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

   14. Applied egg-rr 94.5%

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

       1. *-lft-identity 94.5%

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

       2. associate-/r/ 94.5%

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

       3. associate-/l* 93.4%

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

       4. associate-*r/ 94.5%

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

   16. Simplified 94.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.1 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell} \cdot \left(\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot \left(h \cdot \frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D\_m \cdot \left(0.25 \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\frac{d}{M\_m}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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) -2e-292)
   (*
    w0
    (sqrt
     (- 1.0 (* (/ h l) (* D_m (* 0.25 (/ (* D_m (/ M_m d)) (/ d M_m))))))))
   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) <= -2e-292) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (0.25 * ((D_m * (M_m / d)) / (d / M_m)))))));
	} 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) <= (-2d-292)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (d_m * (0.25d0 * ((d_m * (m_m / d)) / (d / m_m)))))))
    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) <= -2e-292) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (D_m * (0.25 * ((D_m * (M_m / d)) / (d / M_m)))))));
	} 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) <= -2e-292:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (D_m * (0.25 * ((D_m * (M_m / d)) / (d / M_m)))))))
	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) <= -2e-292)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(D_m * Float64(0.25 * Float64(Float64(D_m * Float64(M_m / d)) / Float64(d / M_m))))))));
	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) <= -2e-292)
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (0.25 * ((D_m * (M_m / d)) / (d / M_m)))))));
	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], -2e-292], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(D$95$m * N[(0.25 * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -2.0000000000000001e-292

   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 78.2%

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

      1. unpow2 78.2%

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

      2. associate-/r* 78.2%

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

      3. associate-*r/ 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-/r* 78.2%

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

      6. associate-*r/ 77.1%

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

      7. clear-num 77.0%

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

      8. un-div-inv 77.1%

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

      9. associate-/r/ 77.1%

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

      10. div-inv 77.1%

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

      11. *-commutative 77.1%

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

      12. metadata-eval 77.1%

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

      13. associate-*r* 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 80.5%

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

      2. associate-/l* 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. associate-/l* 78.2%

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

      3. associate-*r* 78.2%

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

   10. Applied egg-rr 78.2%

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

   12. Simplified 78.2%

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

   if -2.0000000000000001e-292 < (/.f64 h l)

   1. Initial program 87.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in D around 0 96.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D \cdot \left(0.25 \cdot \frac{D \cdot \frac{M}{d}}{\frac{d}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D\_m \cdot \left(0.25 \cdot \left(D\_m \cdot \frac{M\_m}{d \cdot \frac{d}{M\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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) -2e-292)
   (*
    w0
    (sqrt
     (- 1.0 (* (/ h l) (* D_m (* 0.25 (* D_m (/ M_m (* d (/ d M_m))))))))))
   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) <= -2e-292) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (0.25 * (D_m * (M_m / (d * (d / M_m)))))))));
	} 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) <= (-2d-292)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (d_m * (0.25d0 * (d_m * (m_m / (d * (d / m_m)))))))))
    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) <= -2e-292) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (D_m * (0.25 * (D_m * (M_m / (d * (d / M_m)))))))));
	} 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) <= -2e-292:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (D_m * (0.25 * (D_m * (M_m / (d * (d / M_m)))))))))
	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) <= -2e-292)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(D_m * Float64(0.25 * Float64(D_m * Float64(M_m / Float64(d * Float64(d / M_m))))))))));
	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) <= -2e-292)
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (0.25 * (D_m * (M_m / (d * (d / M_m)))))))));
	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], -2e-292], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(D$95$m * N[(0.25 * N[(D$95$m * N[(M$95$m / N[(d * N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -2.0000000000000001e-292

   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 78.2%

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

      1. unpow2 78.2%

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

      2. associate-/r* 78.2%

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

      3. associate-*r/ 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-/r* 78.2%

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

      6. associate-*r/ 77.1%

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

      7. clear-num 77.0%

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

      8. un-div-inv 77.1%

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

      9. associate-/r/ 77.1%

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

      10. div-inv 77.1%

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

      11. *-commutative 77.1%

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

      12. metadata-eval 77.1%

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

      13. associate-*r* 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 80.5%

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

      2. associate-/l* 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. associate-/l* 78.2%

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

      3. associate-*r* 78.2%

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

   10. Applied egg-rr 78.2%

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

   12. Simplified 78.2%

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

       1. associate-*l/ 80.5%

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

       2. *-commutative 80.5%

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

       3. associate-/l* 78.1%

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

   14. Applied egg-rr 78.1%

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

       1. associate-*l/ 75.8%

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

       2. associate-/l/ 74.6%

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

   16. Simplified 74.6%

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

   if -2.0000000000000001e-292 < (/.f64 h l)

   1. Initial program 87.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in D around 0 96.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D \cdot \left(0.25 \cdot \left(D \cdot \frac{M}{d \cdot \frac{d}{M}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (/ (* (/ D_m (* d (/ 2.0 M_m))) (* h (* 0.5 (* D_m (/ M_m 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 - (((D_m / (d * (2.0 / M_m))) * (h * (0.5 * (D_m * (M_m / 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 - (((d_m / (d * (2.0d0 / m_m))) * (h * (0.5d0 * (d_m * (m_m / 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 - (((D_m / (d * (2.0 / M_m))) * (h * (0.5 * (D_m * (M_m / 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 - (((D_m / (d * (2.0 / M_m))) * (h * (0.5 * (D_m * (M_m / 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(Float64(D_m / Float64(d * Float64(2.0 / M_m))) * Float64(h * Float64(0.5 * Float64(D_m * Float64(M_m / 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 - (((D_m / (d * (2.0 / M_m))) * (h * (0.5 * (D_m * (M_m / 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[(N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.4%

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

   1. unpow2 81.4%

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

   2. associate-/r* 81.4%

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

   3. associate-*r/ 81.4%

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

   4. *-commutative 81.4%

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

   5. associate-/r* 81.4%

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

   6. associate-*r/ 80.3%

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

   7. clear-num 80.3%

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

   8. un-div-inv 80.3%

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

   9. associate-/r/ 80.3%

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

   10. div-inv 80.3%

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

   11. *-commutative 80.3%

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

   12. metadata-eval 80.3%

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

   13. associate-*r* 80.3%

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

6. Applied egg-rr 80.3%

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

   1. associate-*r/ 85.2%

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

   2. associate-/l* 86.0%

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

8. Applied egg-rr 86.0%

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

   1. pow1 86.0%

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

   2. associate-*l* 87.4%

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

   3. associate-*r* 87.4%

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

10. Applied egg-rr 87.4%

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

    1. unpow1 87.4%

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

    2. associate-*l/ 87.4%

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

    3. associate-/l* 87.4%

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

    4. *-commutative 87.4%

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

    5. *-commutative 87.4%

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

    6. associate-*r/ 87.4%

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

    7. associate-/l* 87.8%

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

12. Simplified 87.8%

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

Alternative 7: 87.2% accurate, 1.8× speedup

Math

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

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 (/ (/ D_m d) 2.0)) (* h (/ (* 0.5 (* D_m (/ M_m 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 * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / 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 * ((d_m / d) / 2.0d0)) * (h * ((0.5d0 * (d_m * (m_m / 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 * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / 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 * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / 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(Float64(D_m / d) / 2.0)) * Float64(h * Float64(Float64(0.5 * Float64(D_m * Float64(M_m / 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 * ((D_m / d) / 2.0)) * (h * ((0.5 * (D_m * (M_m / 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[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.4%

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

   1. unpow2 81.4%

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

   2. associate-/r* 81.4%

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

   3. associate-*r/ 81.4%

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

   4. *-commutative 81.4%

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

   5. associate-/r* 81.4%

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

   6. associate-*r/ 80.3%

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

   7. clear-num 80.3%

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

   8. un-div-inv 80.3%

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

   9. associate-/r/ 80.3%

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

   10. div-inv 80.3%

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

   11. *-commutative 80.3%

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

   12. metadata-eval 80.3%

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

   13. associate-*r* 80.3%

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

6. Applied egg-rr 80.3%

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

   1. associate-*r/ 85.2%

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

   2. associate-/l* 86.0%

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

8. Applied egg-rr 86.0%

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

   1. pow1 86.0%

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

   2. associate-*l* 87.4%

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

   3. associate-*r* 87.4%

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

10. Applied egg-rr 87.4%

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

    1. unpow1 87.4%

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

    2. associate-*l/ 87.4%

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

    3. associate-/l* 87.4%

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

    4. *-commutative 87.4%

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

    5. *-commutative 87.4%

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

    6. associate-*r/ 87.4%

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

    7. associate-/l* 87.8%

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

12. Simplified 87.8%

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

    1. *-un-lft-identity 87.8%

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

    2. associate-/l* 88.5%

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

    3. associate-/r* 87.0%

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

    4. associate-*r/ 88.1%

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

14. Applied egg-rr 88.1%

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

    1. *-lft-identity 88.1%

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

    2. associate-/r/ 88.1%

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

    3. associate-/l* 88.1%

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

    4. associate-*r/ 87.4%

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

16. Simplified 87.4%

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

17. Final simplification 87.4%

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

Alternative 8: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 2.05e+61)
   w0
   (+ w0 (* -0.125 (* (* h (/ w0 l)) (pow (* D_m (/ M_m d)) 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2.05e+61) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * pow((D_m * (M_m / d)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 2.05d+61) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * (w0 / l)) * ((d_m * (m_m / d)) ** 2.0d0)))
    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 (D_m <= 2.05e+61) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * Math.pow((D_m * (M_m / d)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 2.05e+61:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * math.pow((D_m * (M_m / d)), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 2.05e+61)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 / l)) * (Float64(D_m * Float64(M_m / d)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 2.05e+61)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * ((D_m * (M_m / d)) ^ 2.0)));
	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[D$95$m, 2.05e+61], w0, N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.04999999999999986e61

   1. Initial program 81.9%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in D around 0 75.7%

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

   if 2.04999999999999986e61 < D

   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 81.3%

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

   5. Taylor expanded in D around 0 43.0%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative 43.0%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. fma-define 43.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

      3. associate-*r* 41.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}, w0\right) \]

      4. unpow2 41.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      5. unpow2 41.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      6. swap-sqr 59.3%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      7. unpow2 59.3%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

   7. Simplified 59.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 59.3%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. times-frac 60.8%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right)} + w0 \]

   9. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right) + w0} \]

   10. Taylor expanded in D around 0 43.0%

       \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} + w0 \]
   11. Step-by-step derivation

       1. associate-*r* 41.2%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} + w0 \]

       2. unpow2 41.2%

          \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + w0 \]

       3. unpow2 41.2%

          \[\leadsto -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + w0 \]

       4. swap-sqr 59.3%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + w0 \]

       5. unpow2 59.3%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + w0 \]

       6. *-commutative 59.3%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \ell} + w0 \]

       7. *-commutative 59.3%

          \[\leadsto -0.125 \cdot \frac{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}{\color{blue}{\ell \cdot {d}^{2}}} + w0 \]

       8. times-frac 60.8%

          \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{h \cdot w0}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right)} + w0 \]

       9. associate-*r/ 62.5%

          \[\leadsto -0.125 \cdot \left(\color{blue}{\left(h \cdot \frac{w0}{\ell}\right)} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right) + w0 \]

       10. unpow2 62.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}}\right) + w0 \]

       11. unpow2 62.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{d \cdot d}}\right) + w0 \]

       12. times-frac 68.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}\right) + w0 \]

       13. unpow1 68.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \left(\color{blue}{{\left(\frac{D \cdot M}{d}\right)}^{1}} \cdot \frac{D \cdot M}{d}\right)\right) + w0 \]

       14. pow-plus 68.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \color{blue}{{\left(\frac{D \cdot M}{d}\right)}^{\left(1 + 1\right)}}\right) + w0 \]

       15. associate-*r/ 68.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{\left(1 + 1\right)}\right) + w0 \]

       16. metadata-eval 68.5%

           \[\leadsto -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{\color{blue}{2}}\right) + w0 \]

   12. Simplified 68.5%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} + w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.4%

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

   1. unpow2 81.4%

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

   2. associate-/r* 81.4%

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

   3. associate-*r/ 81.4%

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

   4. *-commutative 81.4%

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

   5. associate-/r* 81.4%

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

   6. associate-*r/ 80.3%

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

   7. clear-num 80.3%

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

   8. un-div-inv 80.3%

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

   9. associate-/r/ 80.3%

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

   10. div-inv 80.3%

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

   11. *-commutative 80.3%

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

   12. metadata-eval 80.3%

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

   13. associate-*r* 80.3%

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

6. Applied egg-rr 80.3%

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

   1. associate-*r/ 85.2%

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

   2. associate-/l* 86.0%

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

8. Applied egg-rr 86.0%

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

   1. pow1 86.0%

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

   2. associate-*l* 87.4%

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

   3. associate-*r* 87.4%

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

10. Applied egg-rr 87.4%

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

    1. unpow1 87.4%

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

    2. associate-*l/ 87.4%

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

    3. *-commutative 87.4%

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

    4. *-commutative 87.4%

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

    5. associate-*l/ 87.8%

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

    6. associate-*l* 87.8%

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

    7. associate-*l* 86.0%

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

    8. *-commutative 86.0%

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

    9. associate-*l/ 84.1%

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

    10. associate-*l/ 86.0%

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

    11. associate-*r* 84.9%

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

12. Simplified 84.9%

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

13. Final simplification 84.9%

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

Alternative 10: 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 81.8%

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

3. Simplified 81.4%

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

5. Taylor expanded in D around 0 66.4%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))))