Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.0%

Time: 14.4s

Alternatives: 6

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: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-179}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(M\_m \cdot \frac{D}{d}\right)\right) \cdot \frac{h \cdot \left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right) \cdot \left(0.5 \cdot \left(\left(h \cdot \frac{M\_m}{d}\right) \cdot \frac{D}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= l 5e-179)
   (*
    w0
    (sqrt
     (- 1.0 (* (* 0.5 (* M_m (/ D d))) (/ (* h (* M_m (* 0.5 (/ D d)))) l)))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* D (* M_m (/ 0.5 d))) (* 0.5 (* (* h (/ M_m d)) (/ D l)))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5e-179) {
		tmp = w0 * sqrt((1.0 - ((0.5 * (M_m * (D / d))) * ((h * (M_m * (0.5 * (D / d)))) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D * (M_m * (0.5 / d))) * (0.5 * ((h * (M_m / d)) * (D / l))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5d-179) then
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (m_m * (d / d_1))) * ((h * (m_m * (0.5d0 * (d / d_1)))) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d * (m_m * (0.5d0 / d_1))) * (0.5d0 * ((h * (m_m / d_1)) * (d / l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5e-179) {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (M_m * (D / d))) * ((h * (M_m * (0.5 * (D / d)))) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D * (M_m * (0.5 / d))) * (0.5 * ((h * (M_m / d)) * (D / l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if l <= 5e-179:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (M_m * (D / d))) * ((h * (M_m * (0.5 * (D / d)))) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D * (M_m * (0.5 / d))) * (0.5 * ((h * (M_m / d)) * (D / l))))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (l <= 5e-179)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(M_m * Float64(D / d))) * Float64(Float64(h * Float64(M_m * Float64(0.5 * Float64(D / d)))) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D * Float64(M_m * Float64(0.5 / d))) * Float64(0.5 * Float64(Float64(h * Float64(M_m / d)) * Float64(D / l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (l <= 5e-179)
		tmp = w0 * sqrt((1.0 - ((0.5 * (M_m * (D / d))) * ((h * (M_m * (0.5 * (D / d)))) / l))));
	else
		tmp = w0 * sqrt((1.0 - ((D * (M_m * (0.5 / d))) * (0.5 * ((h * (M_m / d)) * (D / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[l, 5e-179], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999998e-179

   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 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*r/ 81.2%

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

      3. div-inv 81.2%

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

      4. associate-*l* 79.9%

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

      5. associate-/r* 79.9%

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

      6. metadata-eval 79.9%

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

      7. *-un-lft-identity 79.9%

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

      8. times-frac 79.9%

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

      9. metadata-eval 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. associate-*r/ 86.9%

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

      2. associate-*r* 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*r* 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-*l* 88.4%

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

      2. associate-*l* 88.3%

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

   10. Simplified 88.3%

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

       1. associate-/l* 89.0%

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

       2. *-commutative 89.0%

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

       3. associate-*l* 89.0%

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

   12. Applied egg-rr 89.0%

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

   13. Taylor expanded in D around 0 89.0%

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

       1. *-commutative 89.0%

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

       2. associate-/l* 89.6%

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

   15. Simplified 89.6%

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

   if 4.9999999999999998e-179 < l

   1. Initial program 81.6%

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

   3. Simplified 82.5%

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

      1. unpow2 82.5%

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

      2. associate-*r/ 81.7%

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

      3. div-inv 81.7%

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

      4. associate-*l* 78.7%

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

      5. associate-/r* 78.7%

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

      6. metadata-eval 78.7%

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

      7. *-un-lft-identity 78.7%

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

      8. times-frac 78.7%

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

      9. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. associate-*r/ 79.9%

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

      2. associate-*r* 82.8%

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

      3. *-commutative 82.8%

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

      4. associate-*r* 82.8%

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

   8. Applied egg-rr 82.8%

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

      1. associate-*l* 86.6%

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

      2. associate-*l* 83.7%

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

   10. Simplified 83.7%

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

       1. associate-/l* 84.5%

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

       2. *-commutative 84.5%

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

       3. associate-*l* 84.5%

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

   12. Applied egg-rr 84.5%

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

   13. Taylor expanded in h around 0 81.7%

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

       1. *-commutative 81.7%

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

       2. *-commutative 81.7%

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

       3. times-frac 80.6%

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

       4. associate-/l* 84.4%

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

   15. Simplified 84.4%

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

Alternative 2: 83.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -9.5 \cdot 10^{-245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(D \cdot M\_m\right) \cdot \frac{\frac{D \cdot M\_m}{d} \cdot 0.25}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (/ h l) -9.5e-245)
   (*
    w0
    (sqrt (- 1.0 (* (/ h l) (* (* D M_m) (/ (* (/ (* D M_m) d) 0.25) d))))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -9.5e-245) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * M_m) * ((((D * M_m) / d) * 0.25) / d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-9.5d-245)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d * m_m) * ((((d * m_m) / d_1) * 0.25d0) / d_1)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -9.5e-245) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((D * M_m) * ((((D * M_m) / d) * 0.25) / d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (h / l) <= -9.5e-245:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((D * M_m) * ((((D * M_m) / d) * 0.25) / d)))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -9.5e-245)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(D * M_m) * Float64(Float64(Float64(Float64(D * M_m) / d) * 0.25) / d))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -9.5e-245)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * M_m) * ((((D * M_m) / d) * 0.25) / d)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -9.5e-245], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * N[(N[(N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.5000000000000002e-245

   1. Initial program 78.5%

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

   3. Simplified 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r/ 78.5%

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

      3. div-inv 78.5%

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

      4. associate-*l* 78.5%

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

      5. associate-/r* 78.5%

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

      6. metadata-eval 78.5%

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

      7. *-un-lft-identity 78.5%

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

      8. times-frac 78.5%

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

      9. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 79.8%

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

      2. associate-*r* 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-*r* 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. associate-/l* 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*l* 78.5%

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

      4. associate-*r/ 78.5%

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

      5. associate-*r/ 78.5%

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

   10. Simplified 78.5%

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

   11. Taylor expanded in d around 0 78.5%

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

   if -9.5000000000000002e-245 < (/.f64 h l)

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in M around 0 91.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -9.5 \cdot 10^{-245}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{D \cdot M}{d} \cdot 0.25}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - ((d * (m_m * (0.5d0 / d_1))) * ((h * (0.5d0 * ((d * m_m) / d_1))) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Simplified 82.1%

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

   1. unpow2 82.1%

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

   2. associate-*r/ 81.4%

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

   3. div-inv 81.4%

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

   4. associate-*l* 79.4%

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

   5. associate-/r* 79.4%

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

   6. metadata-eval 79.4%

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

   7. *-un-lft-identity 79.4%

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

   8. times-frac 79.4%

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

   9. metadata-eval 79.4%

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

6. Applied egg-rr 79.4%

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

   1. associate-*r/ 84.2%

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

   2. associate-*r* 86.1%

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

   3. *-commutative 86.1%

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

   4. associate-*r* 86.1%

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

8. Applied egg-rr 86.1%

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

   1. associate-*l* 87.7%

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

   2. associate-*l* 86.5%

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

10. Simplified 86.5%

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

    1. associate-/l* 87.2%

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

    2. *-commutative 87.2%

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

    3. associate-*l* 87.2%

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

12. Applied egg-rr 87.2%

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

13. Taylor expanded in M around 0 87.7%

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

Alternative 4: 89.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - ((0.5d0 * (m_m * (d / d_1))) * ((h * (m_m * (0.5d0 * (d / d_1)))) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Simplified 82.1%

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

   1. unpow2 82.1%

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

   2. associate-*r/ 81.4%

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

   3. div-inv 81.4%

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

   4. associate-*l* 79.4%

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

   5. associate-/r* 79.4%

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

   6. metadata-eval 79.4%

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

   7. *-un-lft-identity 79.4%

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

   8. times-frac 79.4%

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

   9. metadata-eval 79.4%

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

6. Applied egg-rr 79.4%

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

   1. associate-*r/ 84.2%

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

   2. associate-*r* 86.1%

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

   3. *-commutative 86.1%

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

   4. associate-*r* 86.1%

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

8. Applied egg-rr 86.1%

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

   1. associate-*l* 87.7%

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

   2. associate-*l* 86.5%

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

10. Simplified 86.5%

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

    1. associate-/l* 87.2%

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

    2. *-commutative 87.2%

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

    3. associate-*l* 87.2%

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

12. Applied egg-rr 87.2%

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

13. Taylor expanded in D around 0 88.0%

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

    1. *-commutative 88.0%

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

    2. associate-/l* 89.1%

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

15. Simplified 89.1%

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

Alternative 5: 84.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - ((h * (d * (m_m * ((((d * m_m) * 0.25d0) / d_1) / d_1)))) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Simplified 82.1%

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

   1. unpow2 82.1%

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

   2. associate-*r/ 81.4%

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

   3. div-inv 81.4%

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

   4. associate-*l* 79.4%

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

   5. associate-/r* 79.4%

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

   6. metadata-eval 79.4%

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

   7. *-un-lft-identity 79.4%

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

   8. times-frac 79.4%

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

   9. metadata-eval 79.4%

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

6. Applied egg-rr 79.4%

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

   1. associate-*r/ 84.2%

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

   2. associate-*r* 86.1%

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

   3. *-commutative 86.1%

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

   4. associate-*r* 86.1%

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

8. Applied egg-rr 86.1%

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

   1. associate-/l* 81.4%

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

   2. *-commutative 81.4%

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

   3. associate-*l* 79.4%

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

   4. associate-*r/ 79.5%

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

   5. associate-*r/ 79.5%

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

10. Simplified 79.5%

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

11. Taylor expanded in d around 0 79.5%

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

    1. associate-*l/ 84.5%

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

    2. associate-*l* 84.5%

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

    3. associate-*r/ 84.5%

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

13. Applied egg-rr 84.5%

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

14. Final simplification 84.5%

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

Alternative 6: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Simplified 82.1%

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

5. Taylor expanded in M around 0 67.1%

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

Reproduce

herbie shell --seed 2024113 
(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))))))