Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 87.6%

Time: 15.2s

Alternatives: 5

Speedup: 1.7×

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 1.7× speedup

Math

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

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
   (+
    (-
     2.0
     (* h (* (* (* D 0.5) (/ M_m d)) (/ 1.0 (/ l (* 0.5 (* D (/ M_m d))))))))
    -1.0))))

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(((2.0 - (h * (((D * 0.5) * (M_m / d)) * (1.0 / (l / (0.5 * (D * (M_m / d)))))))) + -1.0));
}

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(((2.0d0 - (h * (((d * 0.5d0) * (m_m / d_1)) * (1.0d0 / (l / (0.5d0 * (d * (m_m / d_1)))))))) + (-1.0d0)))
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(((2.0 - (h * (((D * 0.5) * (M_m / d)) * (1.0 / (l / (0.5 * (D * (M_m / d)))))))) + -1.0));
}

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(((2.0 - (h * (((D * 0.5) * (M_m / d)) * (1.0 / (l / (0.5 * (D * (M_m / d)))))))) + -1.0))

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(Float64(2.0 - Float64(h * Float64(Float64(Float64(D * 0.5) * Float64(M_m / d)) * Float64(1.0 / Float64(l / Float64(0.5 * Float64(D * Float64(M_m / d)))))))) + -1.0)))
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(((2.0 - (h * (((D * 0.5) * (M_m / d)) * (1.0 / (l / (0.5 * (D * (M_m / d)))))))) + -1.0));
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[(N[(2.0 - N[(h * N[(N[(N[(D * 0.5), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(l / N[(0.5 * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 81.1%

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

   1. *-commutative 81.1%

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

   2. unpow2 81.1%

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

   3. associate-*r* 82.7%

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

   4. *-commutative 82.7%

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

   5. associate-/r* 82.7%

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

   6. associate-*l/ 80.8%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. associate-/r* 81.4%

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

   10. associate-*l/ 82.0%

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

   11. associate-*r/ 83.9%

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

   12. associate-*r* 82.7%

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

   13. unpow2 82.7%

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

   14. *-commutative 82.7%

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

   15. expm1-log1p-u 82.3%

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

6. Applied egg-rr 80.8%

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

   1. expm1-undefine 80.8%

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

   2. sub-neg 80.8%

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

   3. log1p-undefine 80.8%

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

   4. rem-exp-log 81.1%

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

   5. associate-+r- 81.1%

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

   6. metadata-eval 81.1%

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

   7. associate-*l/ 86.1%

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

   8. associate-/l* 87.6%

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

   9. associate-*r/ 88.2%

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

   10. times-frac 87.6%

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

   11. associate-*r/ 88.2%

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

   12. metadata-eval 88.2%

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

8. Simplified 88.2%

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

   1. associate-*r/ 87.6%

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

   2. unpow2 87.6%

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

   3. div-inv 87.6%

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

   4. metadata-eval 87.6%

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

   5. div-inv 87.6%

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

   6. metadata-eval 87.6%

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

10. Applied egg-rr 87.6%

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

    1. associate-/l* 89.6%

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

12. Applied egg-rr 89.6%

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

    1. clear-num 89.6%

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

    2. inv-pow 89.6%

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

14. Applied egg-rr 89.6%

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

    1. unpow-1 89.6%

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

    2. associate-*r/ 87.8%

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

    3. *-commutative 87.8%

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

    4. associate-*r* 87.8%

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

    5. associate-*r/ 87.8%

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

    6. associate-/l* 89.6%

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

16. Simplified 89.6%

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

Alternative 2: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{2 \cdot d}\\ \mathbf{if}\;D \leq 17500000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t\_0 \cdot t\_0\right) \cdot \frac{h}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D (/ M_m (* 2.0 d)))))
   (if (<= D 17500000000000.0)
     w0
     (* w0 (sqrt (- 1.0 (* (* t_0 t_0) (/ h 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 t_0 = D * (M_m / (2.0 * d));
	double tmp;
	if (D <= 17500000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((t_0 * t_0) * (h / 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) :: t_0
    real(8) :: tmp
    t_0 = d * (m_m / (2.0d0 * d_1))
    if (d <= 17500000000000.0d0) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((t_0 * t_0) * (h / 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 t_0 = D * (M_m / (2.0 * d));
	double tmp;
	if (D <= 17500000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * t_0) * (h / 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):
	t_0 = D * (M_m / (2.0 * d))
	tmp = 0
	if D <= 17500000000000.0:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * t_0) * (h / 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)
	t_0 = Float64(D * Float64(M_m / Float64(2.0 * d)))
	tmp = 0.0
	if (D <= 17500000000000.0)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * t_0) * Float64(h / 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)
	t_0 = D * (M_m / (2.0 * d));
	tmp = 0.0;
	if (D <= 17500000000000.0)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((t_0 * t_0) * (h / 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_] := Block[{t$95$0 = N[(D * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 17500000000000.0], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.75e13

   1. Initial program 83.3%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in D around 0 74.8%

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

   if 1.75e13 < D

   1. Initial program 75.5%

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

   3. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-/r* 78.0%

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

      3. associate-*l/ 75.5%

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

      4. associate-*r/ 77.2%

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

      5. clear-num 77.2%

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

      6. un-div-inv 77.1%

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

      7. associate-/l* 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 77.1%

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

   8. Simplified 77.1%

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

      1. unpow2 77.1%

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

      2. associate-/r/ 76.2%

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

      3. associate-/r/ 78.0%

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

   10. Applied egg-rr 78.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 17500000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot \left(D \cdot M\_m\right)\\ \mathbf{if}\;D \leq 1150000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ 0.5 d) (* D M_m))))
   (if (<= D 1150000000000.0)
     w0
     (* w0 (sqrt (- 1.0 (* (/ h l) (* t_0 t_0))))))))

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 t_0 = (0.5 / d) * (D * M_m);
	double tmp;
	if (D <= 1150000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / d_1) * (d * m_m)
    if (d <= 1150000000000.0d0) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (t_0 * t_0))))
    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 t_0 = (0.5 / d) * (D * M_m);
	double tmp;
	if (D <= 1150000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	}
	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):
	t_0 = (0.5 / d) * (D * M_m)
	tmp = 0
	if D <= 1150000000000.0:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (t_0 * t_0))))
	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)
	t_0 = Float64(Float64(0.5 / d) * Float64(D * M_m))
	tmp = 0.0
	if (D <= 1150000000000.0)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	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)
	t_0 = (0.5 / d) * (D * M_m);
	tmp = 0.0;
	if (D <= 1150000000000.0)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	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_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 1150000000000.0], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.15e12

   1. Initial program 83.3%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in D around 0 74.8%

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

   if 1.15e12 < D

   1. Initial program 75.5%

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

   3. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-/r* 78.0%

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

      3. associate-*l/ 75.5%

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

      4. clear-num 75.5%

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

      5. *-commutative 75.5%

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

   6. Applied egg-rr 75.5%

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

      1. associate-/r/ 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in d around 0 75.4%

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

       1. unpow2 75.4%

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

   11. Applied egg-rr 75.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1150000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(D \cdot 0.5\right) \cdot \frac{M\_m}{d}\\ w0 \cdot \sqrt{-1 + \left(2 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)} \end{array} \end{array} \]

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
 (let* ((t_0 (* (* D 0.5) (/ M_m d))))
   (* w0 (sqrt (+ -1.0 (- 2.0 (* h (* t_0 (/ t_0 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 t_0 = (D * 0.5) * (M_m / d);
	return w0 * sqrt((-1.0 + (2.0 - (h * (t_0 * (t_0 / 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
    real(8) :: t_0
    t_0 = (d * 0.5d0) * (m_m / d_1)
    code = w0 * sqrt(((-1.0d0) + (2.0d0 - (h * (t_0 * (t_0 / 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) {
	double t_0 = (D * 0.5) * (M_m / d);
	return w0 * Math.sqrt((-1.0 + (2.0 - (h * (t_0 * (t_0 / 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):
	t_0 = (D * 0.5) * (M_m / d)
	return w0 * math.sqrt((-1.0 + (2.0 - (h * (t_0 * (t_0 / 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)
	t_0 = Float64(Float64(D * 0.5) * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(-1.0 + Float64(2.0 - Float64(h * Float64(t_0 * Float64(t_0 / 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)
	t_0 = (D * 0.5) * (M_m / d);
	tmp = w0 * sqrt((-1.0 + (2.0 - (h * (t_0 * (t_0 / 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_] := Block[{t$95$0 = N[(N[(D * 0.5), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(-1.0 + N[(2.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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 81.1%

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

   1. *-commutative 81.1%

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

   2. unpow2 81.1%

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

   3. associate-*r* 82.7%

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

   4. *-commutative 82.7%

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

   5. associate-/r* 82.7%

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

   6. associate-*l/ 80.8%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. associate-/r* 81.4%

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

   10. associate-*l/ 82.0%

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

   11. associate-*r/ 83.9%

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

   12. associate-*r* 82.7%

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

   13. unpow2 82.7%

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

   14. *-commutative 82.7%

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

   15. expm1-log1p-u 82.3%

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

6. Applied egg-rr 80.8%

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

   1. expm1-undefine 80.8%

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

   2. sub-neg 80.8%

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

   3. log1p-undefine 80.8%

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

   4. rem-exp-log 81.1%

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

   5. associate-+r- 81.1%

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

   6. metadata-eval 81.1%

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

   7. associate-*l/ 86.1%

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

   8. associate-/l* 87.6%

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

   9. associate-*r/ 88.2%

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

   10. times-frac 87.6%

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

   11. associate-*r/ 88.2%

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

   12. metadata-eval 88.2%

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

8. Simplified 88.2%

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

   1. associate-*r/ 87.6%

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

   2. unpow2 87.6%

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

   3. div-inv 87.6%

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

   4. metadata-eval 87.6%

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

   5. div-inv 87.6%

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

   6. metadata-eval 87.6%

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

10. Applied egg-rr 87.6%

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

    1. associate-/l* 89.6%

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

12. Applied egg-rr 89.6%

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

13. Final simplification 89.6%

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

Alternative 5: 67.1% 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 81.6%

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

3. Simplified 81.1%

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

5. Taylor expanded in D around 0 69.9%

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

Reproduce

herbie shell --seed 2024165 
(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))))))