Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 90.1%

Time: 13.6s

Alternatives: 4

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d\_m \leq 2000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m} \cdot h}{\frac{d\_m \cdot \frac{2}{M\_m}}{D\_m} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \left(M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\right)}{\frac{2}{M\_m} \cdot \frac{d\_m}{D\_m}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* 2.0 d_m) 2000000.0)
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       (* (/ (* M_m (* D_m 0.5)) d_m) h)
       (* (/ (* d_m (/ 2.0 M_m)) D_m) l)))))
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       (/ (* h (* M_m (/ (* D_m 0.5) d_m))) (* (/ 2.0 M_m) (/ d_m D_m)))
       l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((2.0 * d_m) <= 2000000.0) {
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (((d_m * (2.0 / M_m)) / D_m) * l))));
	} else {
		tmp = w0 * sqrt((1.0 - (((h * (M_m * ((D_m * 0.5) / d_m))) / ((2.0 / M_m) * (d_m / D_m))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((2.0d0 * d_m_1) <= 2000000.0d0) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * (d_m * 0.5d0)) / d_m_1) * h) / (((d_m_1 * (2.0d0 / m_m)) / d_m) * l))))
    else
        tmp = w0 * sqrt((1.0d0 - (((h * (m_m * ((d_m * 0.5d0) / d_m_1))) / ((2.0d0 / m_m) * (d_m_1 / d_m))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((2.0 * d_m) <= 2000000.0) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (((d_m * (2.0 / M_m)) / D_m) * l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((h * (M_m * ((D_m * 0.5) / d_m))) / ((2.0 / M_m) * (d_m / D_m))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (2.0 * d_m) <= 2000000.0:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (((d_m * (2.0 / M_m)) / D_m) * l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((h * (M_m * ((D_m * 0.5) / d_m))) / ((2.0 / M_m) * (d_m / D_m))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(2.0 * d_m) <= 2000000.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) * h) / Float64(Float64(Float64(d_m * Float64(2.0 / M_m)) / D_m) * l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))) / Float64(Float64(2.0 / M_m) * Float64(d_m / D_m))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((2.0 * d_m) <= 2000000.0)
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (((d_m * (2.0 / M_m)) / D_m) * l))));
	else
		tmp = w0 * sqrt((1.0 - (((h * (M_m * ((D_m * 0.5) / d_m))) / ((2.0 / M_m) * (d_m / D_m))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(2.0 * d$95$m), $MachinePrecision], 2000000.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[(d$95$m * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / M$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 2e6

   1. Initial program 81.0%

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

   3. Simplified 80.6%

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

      1. unpow2 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*l/ 79.9%

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

      4. associate-*r/ 79.6%

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

      5. times-frac 79.9%

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

      6. clear-num 79.9%

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

      7. un-div-inv 79.9%

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

      8. *-commutative 79.9%

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

      9. associate-*l/ 81.0%

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

      10. associate-*r/ 80.0%

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

      11. times-frac 81.0%

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

      12. associate-/l* 80.0%

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

      13. div-inv 80.0%

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

      14. associate-/r* 80.0%

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

      15. metadata-eval 80.0%

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

      16. times-frac 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. frac-times 91.0%

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

      2. associate-*r/ 91.0%

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

      3. associate-*r/ 89.7%

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

      4. associate-*r/ 88.3%

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

   8. Applied egg-rr 88.3%

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

   if 2e6 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 84.9%

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

   3. Simplified 86.1%

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

      1. unpow2 86.1%

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

      2. *-commutative 86.1%

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

      3. associate-*l/ 84.9%

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

      4. associate-*r/ 86.2%

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

      5. times-frac 84.9%

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

      6. clear-num 84.9%

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

      7. un-div-inv 84.9%

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

      8. *-commutative 84.9%

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

      9. associate-*l/ 84.9%

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

      10. associate-*r/ 84.9%

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

      11. times-frac 84.9%

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

      12. associate-/l* 84.9%

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

      13. div-inv 84.9%

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

      14. associate-/r* 84.9%

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

      15. metadata-eval 84.9%

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

      16. times-frac 86.2%

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

   6. Applied egg-rr 86.2%

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

      1. associate-*r/ 90.7%

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

      2. associate-*r/ 90.7%

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

      3. associate-*r/ 89.5%

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

      4. associate-*r/ 89.5%

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

   8. Applied egg-rr 89.5%

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

      1. associate-*l/ 90.9%

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

      2. associate-/l* 93.4%

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

      3. associate-/l* 94.6%

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

   10. Applied egg-rr 94.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 2000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot h}{\frac{d \cdot \frac{2}{M}}{D} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)}{\frac{2}{M} \cdot \frac{d}{D}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.9 \cdot 10^{-268}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \left(D\_m \cdot \frac{0.5}{d\_m}\right)\right) \cdot \frac{\frac{h}{\ell}}{2 \cdot \frac{\frac{d\_m}{D\_m}}{M\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1.9e-268)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* M_m (* D_m (/ 0.5 d_m)))
       (/ (/ h l) (* 2.0 (/ (/ d_m D_m) M_m)))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.9e-268) {
		tmp = w0 * sqrt((1.0 - ((M_m * (D_m * (0.5 / d_m))) * ((h / l) / (2.0 * ((d_m / D_m) / M_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-1.9d-268)) then
        tmp = w0 * sqrt((1.0d0 - ((m_m * (d_m * (0.5d0 / d_m_1))) * ((h / l) / (2.0d0 * ((d_m_1 / d_m) / m_m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.9e-268) {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (D_m * (0.5 / d_m))) * ((h / l) / (2.0 * ((d_m / D_m) / M_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1.9e-268:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (D_m * (0.5 / d_m))) * ((h / l) / (2.0 * ((d_m / D_m) / M_m))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1.9e-268)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) * Float64(Float64(h / l) / Float64(2.0 * Float64(Float64(d_m / D_m) / M_m)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1.9e-268)
		tmp = w0 * sqrt((1.0 - ((M_m * (D_m * (0.5 / d_m))) * ((h / l) / (2.0 * ((d_m / D_m) / M_m))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1.9e-268], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[(2.0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.9000000000000001e-268

   1. Initial program 78.9%

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

   3. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-*l/ 78.9%

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

      4. associate-*r/ 78.5%

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

      5. times-frac 78.9%

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

      6. clear-num 78.9%

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

      7. un-div-inv 78.9%

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

      8. *-commutative 78.9%

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

      9. associate-*l/ 78.9%

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

      10. associate-*r/ 77.8%

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

      11. times-frac 78.9%

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

      12. associate-/l* 77.8%

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

      13. div-inv 77.8%

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

      14. associate-/r* 77.8%

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

      15. metadata-eval 77.8%

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

      16. times-frac 78.6%

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

   6. Applied egg-rr 78.6%

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

      1. frac-times 88.0%

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

      2. associate-*r/ 88.0%

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

      3. associate-*r/ 87.7%

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

      4. associate-*r/ 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. times-frac 78.9%

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

      2. associate-*l/ 81.3%

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

      3. associate-/l* 81.3%

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

      4. associate-/l* 81.5%

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

      5. associate-/l* 81.5%

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

      6. associate-/l* 83.3%

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

      7. associate-*l/ 83.3%

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

      8. associate-/l* 83.3%

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

   10. Simplified 83.3%

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

   if -1.9000000000000001e-268 < (/.f64 h l)

   1. Initial program 87.4%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in D around 0 96.2%

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

Alternative 3: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - ((((m_m * (d_m * 0.5d0)) / d_m_1) * h) / (l * (2.0d0 * (d_m_1 / (m_m * d_m)))))))
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (l * (2.0 * (d_m / (M_m * D_m)))))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) * h) / Float64(l * Float64(2.0 * Float64(d_m / Float64(M_m * D_m))))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d_m) * h) / (l * (2.0 * (d_m / (M_m * D_m)))))));
end

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Simplified 82.1%

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

   1. unpow2 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-*l/ 81.3%

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

   4. associate-*r/ 81.4%

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

   5. times-frac 81.3%

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

   6. clear-num 81.3%

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

   7. un-div-inv 81.3%

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

   8. *-commutative 81.3%

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

   9. associate-*l/ 82.0%

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

   10. associate-*r/ 81.3%

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

   11. times-frac 82.0%

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

   12. associate-/l* 81.3%

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

   13. div-inv 81.3%

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

   14. associate-/r* 81.3%

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

   15. metadata-eval 81.3%

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

   16. times-frac 82.2%

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

6. Applied egg-rr 82.2%

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

   1. frac-times 91.4%

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

   2. associate-*r/ 91.4%

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

   3. associate-*r/ 90.4%

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

   4. associate-*r/ 89.0%

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

8. Applied egg-rr 89.0%

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

9. Taylor expanded in M around 0 91.1%

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

10. Final simplification 91.1%

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

Alternative 4: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Simplified 82.1%

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

5. Taylor expanded in D around 0 63.1%

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

Reproduce

herbie shell --seed 2024182 
(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))))))