Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 86.9%

Time: 15.3s

Alternatives: 11

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: 81.2% 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: 86.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{h}{\frac{d}{\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot 0.25\right)}}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{D\_m}{d} \cdot \left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \left(h \cdot \frac{D\_m}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 1e+111)
   (*
    w0
    (sqrt
     (- 1.0 (/ (/ (/ h (/ d (* (* M_m D_m) (* (* M_m D_m) 0.25)))) l) d))))
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (/ D_m d) (* 0.25 (* M_m M_m))) (* h (/ D_m d))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e+111) {
		tmp = w0 * sqrt((1.0 - (((h / (d / ((M_m * D_m) * ((M_m * D_m) * 0.25)))) / l) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 1d+111) then
        tmp = w0 * sqrt((1.0d0 - (((h / (d / ((m_m * d_m) * ((m_m * d_m) * 0.25d0)))) / l) / d)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m / d) * (0.25d0 * (m_m * m_m))) * (h * (d_m / d))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 1e+111) {
		tmp = w0 * Math.sqrt((1.0 - (((h / (d / ((M_m * D_m) * ((M_m * D_m) * 0.25)))) / l) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 1e+111:
		tmp = w0 * math.sqrt((1.0 - (((h / (d / ((M_m * D_m) * ((M_m * D_m) * 0.25)))) / l) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e+111)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / Float64(d / Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * 0.25)))) / l) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m / d) * Float64(0.25 * Float64(M_m * M_m))) * Float64(h * Float64(D_m / d))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 1e+111)
		tmp = w0 * sqrt((1.0 - (((h / (d / ((M_m * D_m) * ((M_m * D_m) * 0.25)))) / l) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+111], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / N[(d / N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.99999999999999957e110

   1. Initial program 82.1%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2}}{d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d} \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}}{d}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}\right), d\right)\right)\right)\right) \]

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}{d} \cdot h}{\ell}\right), d\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}{d} \cdot h\right), \ell\right), d\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}} \cdot h\right), \ell\right), d\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot h}{\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}}\right), \ell\right), d\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{1} \cdot h}{\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}}\right), \ell\right), d\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{\frac{1}{h}}}{\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}}\right), \ell\right), d\right)\right)\right)\right) \]

      7. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}}\right), \ell\right), d\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \left(\frac{d}{\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}}\right)\right), \ell\right), d\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{1}{4}\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{4}\right)\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\left(M \cdot D\right), \left(\left(M \cdot D\right) \cdot \frac{1}{4}\right)\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\left(M \cdot D\right) \cdot \frac{1}{4}\right)\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(M \cdot D\right), \frac{1}{4}\right)\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

      15. *-lowering-*.f64 86.1%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \frac{1}{4}\right)\right)\right)\right), \ell\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 86.1%

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

   if 9.99999999999999957e110 < (*.f64 M D)

   1. Initial program 63.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right), \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 56.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot h\right)}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{h}{\frac{d}{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d} \cdot \left(0.25 \cdot \left(M \cdot M\right)\right)\right) \cdot \left(h \cdot \frac{D}{d}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+287}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(M\_m \cdot \frac{\frac{D\_m}{\frac{d}{D\_m}}}{\frac{d}{M\_m \cdot h}}\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{D\_m \cdot \left(\frac{D\_m}{d} \cdot \left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -1e+287)
   (*
    w0
    (+ 1.0 (* (/ -0.125 l) (* M_m (/ (/ D_m (/ d D_m)) (/ d (* M_m h)))))))
   (if (<= (/ h l) -2e-323)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ h l) (/ (* D_m (* (/ D_m d) (* 0.25 (* M_m M_m)))) d)))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+287) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (M_m * ((D_m / (d / D_m)) / (d / (M_m * h))))));
	} else if ((h / l) <= -2e-323) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * ((D_m / d) * (0.25 * (M_m * M_m)))) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-1d+287)) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (m_m * ((d_m / (d / d_m)) / (d / (m_m * h))))))
    else if ((h / l) <= (-2d-323)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * ((d_m / d) * (0.25d0 * (m_m * m_m)))) / d))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+287) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (M_m * ((D_m / (d / D_m)) / (d / (M_m * h))))));
	} else if ((h / l) <= -2e-323) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((D_m * ((D_m / d) * (0.25 * (M_m * M_m)))) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -1e+287:
		tmp = w0 * (1.0 + ((-0.125 / l) * (M_m * ((D_m / (d / D_m)) / (d / (M_m * h))))))
	elif (h / l) <= -2e-323:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((D_m * ((D_m / d) * (0.25 * (M_m * M_m)))) / d))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+287)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(M_m * Float64(Float64(D_m / Float64(d / D_m)) / Float64(d / Float64(M_m * h)))))));
	elseif (Float64(h / l) <= -2e-323)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(D_m * Float64(Float64(D_m / d) * Float64(0.25 * Float64(M_m * M_m)))) / d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+287)
		tmp = w0 * (1.0 + ((-0.125 / l) * (M_m * ((D_m / (d / D_m)) / (d / (M_m * h))))));
	elseif ((h / l) <= -2e-323)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * ((D_m / d) * (0.25 * (M_m * M_m)))) / d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+287], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d / N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-323], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m * N[(N[(D$95$m / d), $MachinePrecision] * N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1.0000000000000001e287

   1. Initial program 69.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 64.6%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\left(\frac{D}{d} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{D}{\frac{d}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{D}{\frac{d}{D}} \cdot \frac{1}{\color{blue}{\frac{d}{\left(M \cdot M\right) \cdot h}}}\right)\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{D}{\frac{d}{D}} \cdot \frac{1}{\frac{\frac{d}{h}}{\color{blue}{M \cdot M}}}\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\frac{D}{\frac{d}{D}}}{\color{blue}{\frac{\frac{d}{h}}{M \cdot M}}}\right)\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\frac{D}{\frac{d}{D}}}{\frac{\frac{\frac{d}{h}}{M}}{\color{blue}{M}}}\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\frac{D}{\frac{d}{D}}}{\frac{\frac{d}{h}}{M}} \cdot \color{blue}{M}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{\frac{D}{\frac{d}{D}}}{\frac{\frac{d}{h}}{M}}\right), \color{blue}{M}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{D}{\frac{d}{D}}\right), \left(\frac{\frac{d}{h}}{M}\right)\right), M\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \left(\frac{d}{D}\right)\right), \left(\frac{\frac{d}{h}}{M}\right)\right), M\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\frac{d}{h}}{M}\right)\right), M\right)\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{d}{M \cdot h}\right)\right), M\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(d, \left(M \cdot h\right)\right)\right), M\right)\right)\right)\right) \]

      14. *-lowering-*.f64 74.3%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, h\right)\right)\right), M\right)\right)\right)\right) \]

   7. Applied egg-rr 74.3%

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

   if -1.0000000000000001e287 < (/.f64 h l) < -1.97626e-323

   1. Initial program 81.5%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right), \mathsf{/.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot D}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot D\right), d\right), \mathsf{/.f64}\left(h, \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 68.8%

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

   if -1.97626e-323 < (/.f64 h l)

   1. Initial program 78.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 91.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+287}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{d}{M \cdot h}}\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{D \cdot \left(\frac{D}{d} \cdot \left(0.25 \cdot \left(M \cdot M\right)\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.25}{d}\right)\right) \cdot \frac{h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -2e-323)
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (* M_m D_m) (* (* M_m D_m) (/ 0.25 d))) (/ h l)) d))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-2d-323)) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * d_m) * ((m_m * d_m) * (0.25d0 / d))) * (h / l)) / d)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -2e-323:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-323)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * Float64(0.25 / d))) * Float64(h / l)) / d))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -2e-323)
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-323], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.97626e-323

   1. Initial program 79.7%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2}}{d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d} \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}}{d}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}\right), d\right)\right)\right)\right) \]

   4. Applied egg-rr 70.2%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{\frac{1}{4}}{d}\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{1}{4}}{d}\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      8. /-lowering-/.f64 78.4%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{/.f64}\left(\frac{1}{4}, d\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 78.4%

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

   if -1.97626e-323 < (/.f64 h l)

   1. Initial program 78.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 91.7%

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

Alternative 4: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{-75}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.25}{d}\right)\right) \cdot \frac{h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{D\_m}{d} \cdot \left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \left(h \cdot \frac{D\_m}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 1.15e-75)
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (* M_m D_m) (* (* M_m D_m) (/ 0.25 d))) (/ h l)) d))))
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (/ D_m d) (* 0.25 (* M_m M_m))) (* h (/ D_m d))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.15e-75) {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 1.15d-75) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * d_m) * ((m_m * d_m) * (0.25d0 / d))) * (h / l)) / d)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m / d) * (0.25d0 * (m_m * m_m))) * (h * (d_m / d))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.15e-75) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 1.15e-75:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 1.15e-75)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * Float64(0.25 / d))) * Float64(h / l)) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m / d) * Float64(0.25 * Float64(M_m * M_m))) * Float64(h * Float64(D_m / d))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 1.15e-75)
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) * (0.25 / d))) * (h / l)) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (0.25 * (M_m * M_m))) * (h * (D_m / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 1.15e-75], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.15e-75

   1. Initial program 77.3%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2}}{d}\right) \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}}{d} \cdot \frac{h}{\ell}\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}}{d}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2}\right) \cdot \frac{h}{\ell}\right), d\right)\right)\right)\right) \]

   4. Applied egg-rr 73.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot \left(D \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{\frac{1}{4}}{d}\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{4}}{d}\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{\frac{1}{4}}{d}\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(\frac{\frac{1}{4}}{d}\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

      8. /-lowering-/.f64 79.3%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{/.f64}\left(\frac{1}{4}, d\right)\right)\right), \mathsf{/.f64}\left(h, \ell\right)\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 79.3%

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

   if 1.15e-75 < d

   1. Initial program 81.8%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right), \ell\right)\right)\right)\right) \]

   4. Applied egg-rr 75.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right) \cdot \left(\frac{D}{d} \cdot h\right)}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{-75}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right)\right) \cdot \frac{h}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d} \cdot \left(0.25 \cdot \left(M \cdot M\right)\right)\right) \cdot \left(h \cdot \frac{D}{d}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.4 \cdot 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{D\_m \cdot \frac{M\_m \cdot M\_m}{\frac{d}{h}}}{\frac{d}{D\_m}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 7.4e-125)
   w0
   (*
    w0
    (+ 1.0 (* (/ -0.125 l) (/ (* D_m (/ (* M_m M_m) (/ d h))) (/ d D_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7.4e-125) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((D_m * ((M_m * M_m) / (d / h))) / (d / D_m))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 7.4d-125) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * ((d_m * ((m_m * m_m) / (d / h))) / (d / d_m))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7.4e-125) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((D_m * ((M_m * M_m) / (d / h))) / (d / D_m))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 7.4e-125:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((-0.125 / l) * ((D_m * ((M_m * M_m) / (d / h))) / (d / D_m))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 7.4e-125)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(D_m * Float64(Float64(M_m * M_m) / Float64(d / h))) / Float64(d / D_m)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 7.4e-125)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((-0.125 / l) * ((D_m * ((M_m * M_m) / (d / h))) / (d / D_m))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 7.4e-125], w0, N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.3999999999999998e-125

   1. Initial program 79.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 70.2%

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

   if 7.3999999999999998e-125 < M

   1. Initial program 78.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 65.5%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 65.5%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\left(\frac{D}{d} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{D}{\frac{d}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \color{blue}{\frac{D}{\frac{d}{D}}}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\frac{\left(M \cdot M\right) \cdot h}{d} \cdot D}{\color{blue}{\frac{d}{D}}}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot D\right), \color{blue}{\left(\frac{d}{D}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d}\right), D\right), \left(\frac{\color{blue}{d}}{D}\right)\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{d}{\left(M \cdot M\right) \cdot h}}\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      8. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{d}{h}}{M \cdot M}}\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot M}{\frac{d}{h}}\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), \left(\frac{d}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{d}{h}\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), D\right), \left(\frac{d}{D}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 71.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(d, h\right)\right), D\right), \mathsf{/.f64}\left(d, \color{blue}{D}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 71.8%

      \[\leadsto w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \color{blue}{\frac{\frac{M \cdot M}{\frac{d}{h}} \cdot D}{\frac{d}{D}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.4 \cdot 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{D \cdot \frac{M \cdot M}{\frac{d}{h}}}{\frac{d}{D}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.7 \cdot 10^{-116}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D\_m \cdot \left(\frac{D\_m}{d} \cdot \frac{M\_m \cdot M\_m}{\frac{d}{h}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 1.7e-116)
   w0
   (*
    w0
    (+ 1.0 (* (/ -0.125 l) (* D_m (* (/ D_m d) (/ (* M_m M_m) (/ d h)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.7e-116) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D_m * ((D_m / d) * ((M_m * M_m) / (d / h))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 1.7d-116) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (d_m * ((d_m / d) * ((m_m * m_m) / (d / h))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.7e-116) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (D_m * ((D_m / d) * ((M_m * M_m) / (d / h))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 1.7e-116:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((-0.125 / l) * (D_m * ((D_m / d) * ((M_m * M_m) / (d / h))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 1.7e-116)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(D_m * Float64(Float64(D_m / d) * Float64(Float64(M_m * M_m) / Float64(d / h)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.7e-116)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((-0.125 / l) * (D_m * ((D_m / d) * ((M_m * M_m) / (d / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 1.7e-116], w0, N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(D$95$m * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.69999999999999996e-116

   1. Initial program 79.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 71.0%

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

   if 1.69999999999999996e-116 < M

   1. Initial program 77.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\left(\frac{D}{d} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{D}{\frac{d}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \color{blue}{\frac{D}{\frac{d}{D}}}\right)\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{D}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{D}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{D}{d}\right), \color{blue}{D}\right)\right)\right)\right) \]

   7. Applied egg-rr 70.2%

      \[\leadsto w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{M \cdot M}{\frac{d}{h}}\right) \cdot D\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-116}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(D \cdot \left(\frac{D}{d} \cdot \frac{M \cdot M}{\frac{d}{h}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\frac{D\_m}{d} \cdot \frac{\left(h \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(D\_m \cdot -0.125\right)}{d}}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 4e+17)
   w0
   (* w0 (/ (* (/ D_m d) (/ (* (* h (* M_m M_m)) (* D_m -0.125)) d)) l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e+17) {
		tmp = w0;
	} else {
		tmp = w0 * (((D_m / d) * (((h * (M_m * M_m)) * (D_m * -0.125)) / d)) / l);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 4d+17) then
        tmp = w0
    else
        tmp = w0 * (((d_m / d) * (((h * (m_m * m_m)) * (d_m * (-0.125d0))) / d)) / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e+17) {
		tmp = w0;
	} else {
		tmp = w0 * (((D_m / d) * (((h * (M_m * M_m)) * (D_m * -0.125)) / d)) / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 4e+17:
		tmp = w0
	else:
		tmp = w0 * (((D_m / d) * (((h * (M_m * M_m)) * (D_m * -0.125)) / d)) / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 4e+17)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(D_m / d) * Float64(Float64(Float64(h * Float64(M_m * M_m)) * Float64(D_m * -0.125)) / d)) / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 4e+17)
		tmp = w0;
	else
		tmp = w0 * (((D_m / d) * (((h * (M_m * M_m)) * (D_m * -0.125)) / d)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 4e+17], w0, N[(w0 * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4e17

   1. Initial program 80.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.5%

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

   if 4e17 < M

   1. Initial program 73.3%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 48.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 48.8%

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

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right), \ell\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(D \cdot D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(d \cdot d\right)\right), \ell\right)\right) \]

      16. *-lowering-*.f64 25.2%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(d, d\right)\right), \ell\right)\right) \]

   8. Simplified 25.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\left(\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D\right) \cdot D}{d \cdot d}\right), \ell\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D}{d} \cdot \frac{D}{d}\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D}{d}\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{-1}{8} \cdot D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{-1}{8} \cdot D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), d\right), \left(\frac{D}{d}\right)\right), \ell\right)\right) \]

      12. /-lowering-/.f64 31.7%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), d\right), \mathsf{/.f64}\left(D, d\right)\right), \ell\right)\right) \]

   10. Applied egg-rr 31.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\frac{D}{d} \cdot \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot -0.125\right)}{d}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(h \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{-0.125}{d}\right) \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 1.5e+20)
   w0
   (* w0 (* (* (* h (* M_m M_m)) (/ -0.125 d)) (/ (/ (* D_m D_m) d) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.5e+20) {
		tmp = w0;
	} else {
		tmp = w0 * (((h * (M_m * M_m)) * (-0.125 / d)) * (((D_m * D_m) / d) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 1.5d+20) then
        tmp = w0
    else
        tmp = w0 * (((h * (m_m * m_m)) * ((-0.125d0) / d)) * (((d_m * d_m) / d) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.5e+20) {
		tmp = w0;
	} else {
		tmp = w0 * (((h * (M_m * M_m)) * (-0.125 / d)) * (((D_m * D_m) / d) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 1.5e+20:
		tmp = w0
	else:
		tmp = w0 * (((h * (M_m * M_m)) * (-0.125 / d)) * (((D_m * D_m) / d) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 1.5e+20)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(h * Float64(M_m * M_m)) * Float64(-0.125 / d)) * Float64(Float64(Float64(D_m * D_m) / d) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.5e+20)
		tmp = w0;
	else
		tmp = w0 * (((h * (M_m * M_m)) * (-0.125 / d)) * (((D_m * D_m) / d) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 1.5e+20], w0, N[(w0 * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.5e20

   1. Initial program 80.6%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.5%

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

   if 1.5e20 < M

   1. Initial program 73.3%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 48.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 48.8%

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

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right), \ell\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(D \cdot D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(d \cdot d\right)\right), \ell\right)\right) \]

      16. *-lowering-*.f64 25.2%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(d, d\right)\right), \ell\right)\right) \]

   8. Simplified 25.2%

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

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d} \cdot \frac{D \cdot D}{d}}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d} \cdot \color{blue}{\frac{\frac{D \cdot D}{d}}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}\right), \color{blue}{\left(\frac{\frac{D \cdot D}{d}}{\ell}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d}\right), \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{\ell}\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{-1}{8}}{d}\right), \left(\frac{\color{blue}{\frac{D \cdot D}{d}}}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{\frac{-1}{8}}{d}\right)\right), \left(\frac{\color{blue}{\frac{D \cdot D}{d}}}{\ell}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right), \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right), \left(\frac{\frac{\color{blue}{D \cdot D}}{d}}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{\frac{-1}{8}}{d}\right)\right), \left(\frac{\frac{D \cdot \color{blue}{D}}{d}}{\ell}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, d\right)\right), \left(\frac{\frac{D \cdot D}{\color{blue}{d}}}{\ell}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, d\right)\right), \mathsf{/.f64}\left(\left(\frac{D \cdot D}{d}\right), \color{blue}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, d\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \ell\right)\right)\right) \]

      13. *-lowering-*.f64 29.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, d\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 29.6%

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

Alternative 9: 71.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.4 \cdot 10^{+96}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(h \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(D\_m \cdot -0.125\right)\right) \cdot \frac{D\_m}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 3.4e+96)
   w0
   (* w0 (* (* (* h (* M_m M_m)) (* D_m -0.125)) (/ D_m (* d (* d l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.4e+96) {
		tmp = w0;
	} else {
		tmp = w0 * (((h * (M_m * M_m)) * (D_m * -0.125)) * (D_m / (d * (d * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 3.4d+96) then
        tmp = w0
    else
        tmp = w0 * (((h * (m_m * m_m)) * (d_m * (-0.125d0))) * (d_m / (d * (d * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.4e+96) {
		tmp = w0;
	} else {
		tmp = w0 * (((h * (M_m * M_m)) * (D_m * -0.125)) * (D_m / (d * (d * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 3.4e+96:
		tmp = w0
	else:
		tmp = w0 * (((h * (M_m * M_m)) * (D_m * -0.125)) * (D_m / (d * (d * l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 3.4e+96)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(h * Float64(M_m * M_m)) * Float64(D_m * -0.125)) * Float64(D_m / Float64(d * Float64(d * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 3.4e+96)
		tmp = w0;
	else
		tmp = w0 * (((h * (M_m * M_m)) * (D_m * -0.125)) * (D_m / (d * (d * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 3.4e+96], w0, N[(w0 * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.4000000000000001e96

   1. Initial program 79.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

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

   if 3.4000000000000001e96 < M

   1. Initial program 75.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 41.4%

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

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right), \ell\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(D \cdot D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(d \cdot d\right)\right), \ell\right)\right) \]

      16. *-lowering-*.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(d, d\right)\right), \ell\right)\right) \]

   8. Simplified 27.0%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D\right) \cdot D}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D\right) \cdot \color{blue}{\frac{D}{d \cdot \left(d \cdot \ell\right)}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot D\right), \color{blue}{\left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right), \left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{-1}{8} \cdot D\right)\right), \left(\frac{\color{blue}{D}}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{-1}{8} \cdot D\right)\right), \left(\frac{\color{blue}{D}}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(h \cdot \left(M \cdot M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), \left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), \left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \left(\frac{-1}{8} \cdot D\right)\right), \left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), \left(\frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), \mathsf{/.f64}\left(D, \color{blue}{\left(d \cdot \left(d \cdot \ell\right)\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 32.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 32.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.4 \cdot 10^{+96}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot -0.125\right)\right) \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3 \cdot 10^{+96}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot h\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 3e+96)
   w0
   (* w0 (* D_m (* (* (* M_m D_m) (* M_m h)) (/ -0.125 (* d (* d l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+96) {
		tmp = w0;
	} else {
		tmp = w0 * (D_m * (((M_m * D_m) * (M_m * h)) * (-0.125 / (d * (d * l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 3d+96) then
        tmp = w0
    else
        tmp = w0 * (d_m * (((m_m * d_m) * (m_m * h)) * ((-0.125d0) / (d * (d * l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+96) {
		tmp = w0;
	} else {
		tmp = w0 * (D_m * (((M_m * D_m) * (M_m * h)) * (-0.125 / (d * (d * l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 3e+96:
		tmp = w0
	else:
		tmp = w0 * (D_m * (((M_m * D_m) * (M_m * h)) * (-0.125 / (d * (d * l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 3e+96)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(D_m * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * h)) * Float64(-0.125 / Float64(d * Float64(d * l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 3e+96)
		tmp = w0;
	else
		tmp = w0 * (D_m * (((M_m * D_m) * (M_m * h)) * (-0.125 / (d * (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 3e+96], w0, N[(w0 * N[(D$95$m * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3e96

   1. Initial program 79.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

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

   if 3e96 < M

   1. Initial program 75.0%

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

   3. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right)\right)\right)\right) \]

   5. Simplified 41.4%

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

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right), \ell\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(D \cdot D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2}\right)\right), \ell\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left(d \cdot d\right)\right), \ell\right)\right) \]

      16. *-lowering-*.f64 27.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(d, d\right)\right), \ell\right)\right) \]

   8. Simplified 27.0%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right)}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \frac{-1}{8}}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \color{blue}{\frac{\frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)}}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \frac{\frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)}\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot \frac{\color{blue}{\frac{-1}{8}}}{d \cdot \left(d \cdot \ell\right)}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(D \cdot \color{blue}{\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \frac{\frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \color{blue}{\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \frac{\frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), \color{blue}{\left(\frac{\frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)}\right)}\right)\right)\right) \]

   10. Applied egg-rr 32.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+96}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot h\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in M around 0

   \[\leadsto \color{blue}{w0} \]
4. Step-by-step derivation

5. Simplified 66.5%

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

Reproduce

herbie shell --seed 2024191 
(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))))))