Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.0%

Time: 18.0s

Alternatives: 14

Speedup: 1.6×

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: 81.0% 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: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{\frac{2 \cdot d}{M\_m}}{D\_m}\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{h}{\ell}}{t\_0}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m \cdot \left(-0.25 \cdot \frac{\frac{D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)}{d}}{\ell}\right)}{d}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (/ (* 2.0 d) M_m) D_m)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) 0.01)
     (* w0 (sqrt (- 1.0 (/ (/ (/ h l) t_0) t_0))))
     (*
      w0
      (sqrt
       (+
        1.0
        (/ (* D_m (* -0.25 (/ (/ (* D_m (* h (* M_m M_m))) d) l))) d)))))))

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 t_0 = ((2.0 * d) / M_m) / D_m;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		tmp = w0 * sqrt((1.0 - (((h / l) / t_0) / t_0)));
	} else {
		tmp = w0 * sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((2.0d0 * d) / m_m) / d_m
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= 0.01d0) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) / t_0) / t_0)))
    else
        tmp = w0 * sqrt((1.0d0 + ((d_m * ((-0.25d0) * (((d_m * (h * (m_m * m_m))) / d) / l))) / d)))
    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 t_0 = ((2.0 * d) / M_m) / D_m;
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) / t_0) / t_0)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	}
	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):
	t_0 = ((2.0 * d) / M_m) / D_m
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= 0.01:
		tmp = w0 * math.sqrt((1.0 - (((h / l) / t_0) / t_0)))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)))
	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)
	t_0 = Float64(Float64(Float64(2.0 * d) / M_m) / D_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) / t_0) / t_0))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m * Float64(-0.25 * Float64(Float64(Float64(D_m * Float64(h * Float64(M_m * M_m))) / d) / l))) / d))));
	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)
	t_0 = ((2.0 * d) / M_m) / D_m;
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= 0.01)
		tmp = w0 * sqrt((1.0 - (((h / l) / t_0) / t_0)));
	else
		tmp = w0 * sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	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_] := Block[{t$95$0 = N[(N[(N[(2.0 * d), $MachinePrecision] / M$95$m), $MachinePrecision] / D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m * N[(-0.25 * N[(N[(N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0100000000000000002

   1. Initial program 89.7%

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. associate-/r* N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. associate-/r* N/A

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

      17. /-lowering-/.f64 N/A

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

      18. /-lowering-/.f64 N/A

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 91.3%

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

   4. Applied egg-rr 91.3%

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

   if 0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 4.1%

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

   6. Applied egg-rr 4.1%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{\frac{h}{\ell}}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}} \cdot D}}{d}} \]

   7. Taylor expanded in h around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\frac{D \cdot \left({M}^{2} \cdot h\right)}{d \cdot \ell}\right)\right), D\right), d\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\frac{\frac{D \cdot \left({M}^{2} \cdot h\right)}{d}}{\ell}\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\left(\frac{D \cdot \left({M}^{2} \cdot h\right)}{d}\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left({M}^{2} \cdot h\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot {M}^{2}\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 64.6%

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

   9. Simplified 64.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{h}{\ell}}{\frac{\frac{2 \cdot d}{M}}{D}}}{\frac{\frac{2 \cdot d}{M}}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot \left(-0.25 \cdot \frac{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\ell}\right)}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{\frac{M\_m}{\frac{d}{M\_m \cdot D\_m}}}}}{d} \cdot \frac{D\_m}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-209}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m}{d} \cdot \left(\left(M\_m \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M\_m}{\frac{d}{D\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m \cdot \left(-0.25 \cdot \frac{\frac{D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)}{d}}{\ell}\right)}{d}}\\ \end{array} \end{array} \]

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) -5e+300)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (/ h (/ -4.0 (/ M_m (/ d (* M_m D_m))))) d) (/ D_m l)))))
   (if (<= (/ h l) -1e-209)
     (*
      w0
      (sqrt
       (+ 1.0 (* (/ D_m d) (* (* M_m (/ (/ h l) -4.0)) (/ M_m (/ d D_m)))))))
     (*
      w0
      (sqrt
       (+
        1.0
        (/ (* D_m (* -0.25 (/ (/ (* D_m (* h (* M_m M_m))) d) l))) d)))))))

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) <= -5e+300) {
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	} else if ((h / l) <= -1e-209) {
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0 * sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	}
	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) <= (-5d+300)) then
        tmp = w0 * sqrt((1.0d0 + (((h / ((-4.0d0) / (m_m / (d / (m_m * d_m))))) / d) * (d_m / l))))
    else if ((h / l) <= (-1d-209)) then
        tmp = w0 * sqrt((1.0d0 + ((d_m / d) * ((m_m * ((h / l) / (-4.0d0))) * (m_m / (d / d_m))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((d_m * ((-0.25d0) * (((d_m * (h * (m_m * m_m))) / d) / l))) / d)))
    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) <= -5e+300) {
		tmp = w0 * Math.sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	} else if ((h / l) <= -1e-209) {
		tmp = w0 * Math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	}
	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) <= -5e+300:
		tmp = w0 * math.sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))))
	elif (h / l) <= -1e-209:
		tmp = w0 * math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)))
	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) <= -5e+300)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / Float64(-4.0 / Float64(M_m / Float64(d / Float64(M_m * D_m))))) / d) * Float64(D_m / l)))));
	elseif (Float64(h / l) <= -1e-209)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m / d) * Float64(Float64(M_m * Float64(Float64(h / l) / -4.0)) * Float64(M_m / Float64(d / D_m)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m * Float64(-0.25 * Float64(Float64(Float64(D_m * Float64(h * Float64(M_m * M_m))) / d) / l))) / d))));
	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) <= -5e+300)
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	elseif ((h / l) <= -1e-209)
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	else
		tmp = w0 * sqrt((1.0 + ((D_m * (-0.25 * (((D_m * (h * (M_m * M_m))) / d) / l))) / d)));
	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], -5e+300], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / N[(-4.0 / N[(M$95$m / N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-209], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * N[(N[(h / l), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m * N[(-0.25 * N[(N[(N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -5.00000000000000026e300

   1. Initial program 32.8%

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

   3. Simplified 32.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

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

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

   6. Applied egg-rr 84.9%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{d} \cdot \frac{D}{\ell}}} \]

   if -5.00000000000000026e300 < (/.f64 h l) < -1e-209

   1. Initial program 81.8%

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

   3. Simplified 71.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l/ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-/r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. associate-/r/ N/A

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

      16. div-inv N/A

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

      17. associate-/r* N/A

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

      18. clear-num N/A

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

   8. Applied egg-rr 80.8%

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

   if -1e-209 < (/.f64 h l)

   1. Initial program 88.5%

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

   3. Simplified 83.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 88.6%

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

   6. Applied egg-rr 88.6%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{\frac{h}{\ell}}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}} \cdot D}}{d}} \]

   7. Taylor expanded in h around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\frac{D \cdot \left({M}^{2} \cdot h\right)}{d \cdot \ell}\right)\right), D\right), d\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\frac{\frac{D \cdot \left({M}^{2} \cdot h\right)}{d}}{\ell}\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\left(\frac{D \cdot \left({M}^{2} \cdot h\right)}{d}\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left({M}^{2} \cdot h\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot {M}^{2}\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), d\right), \ell\right)\right), D\right), d\right)\right)\right)\right) \]

      8. unpow2 N/A

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

      9. *-lowering-*.f64 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{M \cdot D}}}}}{d} \cdot \frac{D}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-209}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \left(\left(M \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot \left(-0.25 \cdot \frac{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}}{\ell}\right)}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{\frac{M\_m}{\frac{d}{M\_m \cdot D\_m}}}}}{d} \cdot \frac{D\_m}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-275}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m}{d} \cdot \left(\left(M\_m \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M\_m}{\frac{d}{D\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e+300)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (/ h (/ -4.0 (/ M_m (/ d (* M_m D_m))))) d) (/ D_m l)))))
   (if (<= (/ h l) -1e-275)
     (*
      w0
      (sqrt
       (+ 1.0 (* (/ D_m d) (* (* M_m (/ (/ h l) -4.0)) (/ M_m (/ d D_m)))))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e+300) {
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	} else if ((h / l) <= -1e-275) {
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-5d+300)) then
        tmp = w0 * sqrt((1.0d0 + (((h / ((-4.0d0) / (m_m / (d / (m_m * d_m))))) / d) * (d_m / l))))
    else if ((h / l) <= (-1d-275)) then
        tmp = w0 * sqrt((1.0d0 + ((d_m / d) * ((m_m * ((h / l) / (-4.0d0))) * (m_m / (d / d_m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e+300) {
		tmp = w0 * Math.sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	} else if ((h / l) <= -1e-275) {
		tmp = w0 * Math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e+300:
		tmp = w0 * math.sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))))
	elif (h / l) <= -1e-275:
		tmp = w0 * math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e+300)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / Float64(-4.0 / Float64(M_m / Float64(d / Float64(M_m * D_m))))) / d) * Float64(D_m / l)))));
	elseif (Float64(h / l) <= -1e-275)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m / d) * Float64(Float64(M_m * Float64(Float64(h / l) / -4.0)) * Float64(M_m / Float64(d / D_m)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e+300)
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 / (M_m / (d / (M_m * D_m))))) / d) * (D_m / l))));
	elseif ((h / l) <= -1e-275)
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e+300], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / N[(-4.0 / N[(M$95$m / N[(d / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-275], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * N[(N[(h / l), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -5.00000000000000026e300

   1. Initial program 32.8%

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

   3. Simplified 32.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

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

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

   6. Applied egg-rr 84.9%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{d} \cdot \frac{D}{\ell}}} \]

   if -5.00000000000000026e300 < (/.f64 h l) < -9.99999999999999934e-276

   1. Initial program 82.9%

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

   3. Simplified 72.2%

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

      1. associate-*l/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l/ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-/r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. associate-/r/ N/A

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

      16. div-inv N/A

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

      17. associate-/r* N/A

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

      18. clear-num N/A

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

   8. Applied egg-rr 82.0%

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

   if -9.99999999999999934e-276 < (/.f64 h l)

   1. Initial program 87.8%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 93.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{M \cdot D}}}}}{d} \cdot \frac{D}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-275}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \left(\left(M \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D\_m \cdot D\_m}{\frac{\ell \cdot \left(d \cdot d\right)}{M\_m \cdot \left(M\_m \cdot \left(h \cdot -0.125\right)\right)}}\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-275}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m}{d} \cdot \left(\left(M\_m \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M\_m}{\frac{d}{D\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (*
    w0
    (+ 1.0 (/ (* D_m D_m) (/ (* l (* d d)) (* M_m (* M_m (* h -0.125)))))))
   (if (<= (/ h l) -1e-275)
     (*
      w0
      (sqrt
       (+ 1.0 (* (/ D_m d) (* (* M_m (/ (/ h l) -4.0)) (/ M_m (/ d D_m)))))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * (1.0 + ((D_m * D_m) / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	} else if ((h / l) <= -1e-275) {
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * (1.0 + ((D_m * D_m) / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	} else if ((h / l) <= -1e-275) {
		tmp = w0 * Math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * (1.0 + ((D_m * D_m) / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))))
	elif (h / l) <= -1e-275:
		tmp = w0 * math.sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D_m * D_m) / Float64(Float64(l * Float64(d * d)) / Float64(M_m * Float64(M_m * Float64(h * -0.125)))))));
	elseif (Float64(h / l) <= -1e-275)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m / d) * Float64(Float64(M_m * Float64(Float64(h / l) / -4.0)) * Float64(M_m / Float64(d / D_m)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * (1.0 + ((D_m * D_m) / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	elseif ((h / l) <= -1e-275)
		tmp = w0 * sqrt((1.0 + ((D_m / d) * ((M_m * ((h / l) / -4.0)) * (M_m / (d / D_m))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[(1.0 + N[(N[(D$95$m * D$95$m), $MachinePrecision] / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(M$95$m * N[(M$95$m * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-275], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * N[(N[(h / l), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 27.9%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 67.5%

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

   7. Simplified 67.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 72.3%

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

   9. Applied egg-rr 72.3%

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

       1. *-commutative N/A

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

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

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

   11. Applied egg-rr 67.5%

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

   if -inf.0 < (/.f64 h l) < -9.99999999999999934e-276

   1. Initial program 82.5%

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

   3. Simplified 72.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l/ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/l* N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-/r/ N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l/ N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. associate-/r/ N/A

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

      16. div-inv N/A

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

      17. associate-/r* N/A

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

      18. clear-num N/A

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

   8. Applied egg-rr 81.7%

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

   if -9.99999999999999934e-276 < (/.f64 h l)

   1. Initial program 87.8%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 93.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}}\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-275}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \left(\left(M \cdot \frac{\frac{h}{\ell}}{-4}\right) \cdot \frac{M}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot \left(h \cdot \frac{D\_m}{\ell}\right)\\ \mathbf{if}\;M\_m \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M\_m \leq 1.75 \cdot 10^{+42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{t\_0}{\frac{d}{\left(M\_m \cdot M\_m\right) \cdot -0.125}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0 \cdot \frac{M\_m}{\frac{d \cdot -4}{M\_m}}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ D_m d) (* h (/ D_m l)))))
   (if (<= M_m 1.65e-85)
     w0
     (if (<= M_m 1.75e+42)
       (* w0 (+ 1.0 (/ t_0 (/ d (* (* M_m M_m) -0.125)))))
       (* w0 (sqrt (* t_0 (/ M_m (/ (* d -4.0) M_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 t_0 = (D_m / d) * (h * (D_m / l));
	double tmp;
	if (M_m <= 1.65e-85) {
		tmp = w0;
	} else if (M_m <= 1.75e+42) {
		tmp = w0 * (1.0 + (t_0 / (d / ((M_m * M_m) * -0.125))));
	} else {
		tmp = w0 * sqrt((t_0 * (M_m / ((d * -4.0) / M_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) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d) * (h * (d_m / l))
    if (m_m <= 1.65d-85) then
        tmp = w0
    else if (m_m <= 1.75d+42) then
        tmp = w0 * (1.0d0 + (t_0 / (d / ((m_m * m_m) * (-0.125d0)))))
    else
        tmp = w0 * sqrt((t_0 * (m_m / ((d * (-4.0d0)) / m_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 t_0 = (D_m / d) * (h * (D_m / l));
	double tmp;
	if (M_m <= 1.65e-85) {
		tmp = w0;
	} else if (M_m <= 1.75e+42) {
		tmp = w0 * (1.0 + (t_0 / (d / ((M_m * M_m) * -0.125))));
	} else {
		tmp = w0 * Math.sqrt((t_0 * (M_m / ((d * -4.0) / M_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):
	t_0 = (D_m / d) * (h * (D_m / l))
	tmp = 0
	if M_m <= 1.65e-85:
		tmp = w0
	elif M_m <= 1.75e+42:
		tmp = w0 * (1.0 + (t_0 / (d / ((M_m * M_m) * -0.125))))
	else:
		tmp = w0 * math.sqrt((t_0 * (M_m / ((d * -4.0) / M_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)
	t_0 = Float64(Float64(D_m / d) * Float64(h * Float64(D_m / l)))
	tmp = 0.0
	if (M_m <= 1.65e-85)
		tmp = w0;
	elseif (M_m <= 1.75e+42)
		tmp = Float64(w0 * Float64(1.0 + Float64(t_0 / Float64(d / Float64(Float64(M_m * M_m) * -0.125)))));
	else
		tmp = Float64(w0 * sqrt(Float64(t_0 * Float64(M_m / Float64(Float64(d * -4.0) / M_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)
	t_0 = (D_m / d) * (h * (D_m / l));
	tmp = 0.0;
	if (M_m <= 1.65e-85)
		tmp = w0;
	elseif (M_m <= 1.75e+42)
		tmp = w0 * (1.0 + (t_0 / (d / ((M_m * M_m) * -0.125))));
	else
		tmp = w0 * sqrt((t_0 * (M_m / ((d * -4.0) / M_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_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * N[(h * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.65e-85], w0, If[LessEqual[M$95$m, 1.75e+42], N[(w0 * N[(1.0 + N[(t$95$0 / N[(d / N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(M$95$m / N[(N[(d * -4.0), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.64999999999999986e-85

   1. Initial program 80.0%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.4%

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

   if 1.64999999999999986e-85 < M < 1.75000000000000012e42

   1. Initial program 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 86.3%

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

   9. Applied egg-rr 86.3%

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

       1. associate-*r* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-commutative N/A

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

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

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

       14. *-lowering-*.f64 82.9%

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

   11. Applied egg-rr 82.9%

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

   if 1.75000000000000012e42 < M

   1. Initial program 77.6%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({M}^{2} \cdot \left(h \cdot {D}^{2}\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({M}^{2} \cdot \left({D}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({D}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({D}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({D}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(\left({D}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(\left(D \cdot D\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 21.9%

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

   7. Simplified 21.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 21.6%

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

   9. Applied egg-rr 21.6%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. associate-*l/ N/A

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

       6. associate-/r* N/A

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

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

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. div-inv N/A

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

       14. clear-num N/A

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

       15. un-div-inv N/A

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

       16. /-lowering-/.f64 N/A

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

       17. /-lowering-/.f64 N/A

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

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

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

       19. times-frac N/A

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

       20. associate-*l* N/A

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

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

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

   11. Applied egg-rr 32.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.75 \cdot 10^{+42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\frac{D}{d} \cdot \left(h \cdot \frac{D}{\ell}\right)}{\frac{d}{\left(M \cdot M\right) \cdot -0.125}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\frac{D}{d} \cdot \left(h \cdot \frac{D}{\ell}\right)\right) \cdot \frac{M}{\frac{d \cdot -4}{M}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 8 \cdot 10^{-86}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\frac{D\_m}{d} \cdot \left(h \cdot \frac{D\_m}{\ell}\right)}{\frac{d}{\left(M\_m \cdot M\_m\right) \cdot -0.125}}\right)\\ \end{array} \end{array} \]

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 8e-86)
   w0
   (*
    w0
    (+ 1.0 (/ (* (/ D_m d) (* h (/ D_m l))) (/ d (* (* M_m M_m) -0.125)))))))

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 <= 8e-86) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D_m / d) * (h * (D_m / l))) / (d / ((M_m * M_m) * -0.125))));
	}
	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 <= 8d-86) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((d_m / d) * (h * (d_m / l))) / (d / ((m_m * m_m) * (-0.125d0)))))
    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 <= 8e-86) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D_m / d) * (h * (D_m / l))) / (d / ((M_m * M_m) * -0.125))));
	}
	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 <= 8e-86:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (((D_m / d) * (h * (D_m / l))) / (d / ((M_m * M_m) * -0.125))))
	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 <= 8e-86)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D_m / d) * Float64(h * Float64(D_m / l))) / Float64(d / Float64(Float64(M_m * M_m) * -0.125)))));
	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 <= 8e-86)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (((D_m / d) * (h * (D_m / l))) / (d / ((M_m * M_m) * -0.125))));
	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, 8e-86], w0, N[(w0 * N[(1.0 + N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(h * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.00000000000000068e-86

   1. Initial program 80.0%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.4%

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

   if 8.00000000000000068e-86 < M

   1. Initial program 80.6%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 49.3%

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

   7. Simplified 49.3%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 55.7%

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

   9. Applied egg-rr 55.7%

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

       1. associate-*r* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-commutative N/A

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

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

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

       14. *-lowering-*.f64 58.5%

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

   11. Applied egg-rr 58.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8 \cdot 10^{-86}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\frac{D}{d} \cdot \left(h \cdot \frac{D}{\ell}\right)}{\frac{d}{\left(M \cdot M\right) \cdot -0.125}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 3.55:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D\_m \cdot \frac{D\_m \cdot \left(h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

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 (<= D_m 3.55)
   w0
   (*
    w0
    (+ 1.0 (* D_m (/ (* D_m (* h (* (* M_m M_m) -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 (D_m <= 3.55) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (D_m * ((D_m * (h * ((M_m * M_m) * -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 (d_m <= 3.55d0) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (d_m * ((d_m * (h * ((m_m * m_m) * (-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 (D_m <= 3.55) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (D_m * ((D_m * (h * ((M_m * M_m) * -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 D_m <= 3.55:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (D_m * ((D_m * (h * ((M_m * M_m) * -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 (D_m <= 3.55)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(D_m * Float64(Float64(D_m * Float64(h * Float64(Float64(M_m * M_m) * -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 (D_m <= 3.55)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (D_m * ((D_m * (h * ((M_m * M_m) * -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[D$95$m, 3.55], w0, N[(w0 * N[(1.0 + N[(D$95$m * N[(N[(D$95$m * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.5499999999999998

   1. Initial program 82.5%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 77.0%

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

   if 3.5499999999999998 < D

   1. Initial program 72.7%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 48.4%

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

   7. Simplified 48.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 63.7%

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

   9. Applied egg-rr 63.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.55:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D\_m \cdot D\_m}{d \cdot \ell} \cdot \left(h \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 5.4e-49)
   (*
    w0
    (+ 1.0 (* (/ (* D_m D_m) (* d l)) (* h (/ (* (* M_m M_m) -0.125) 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 (d <= 5.4e-49) {
		tmp = w0 * (1.0 + (((D_m * D_m) / (d * l)) * (h * (((M_m * M_m) * -0.125) / 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 (d <= 5.4d-49) then
        tmp = w0 * (1.0d0 + (((d_m * d_m) / (d * l)) * (h * (((m_m * m_m) * (-0.125d0)) / 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 (d <= 5.4e-49) {
		tmp = w0 * (1.0 + (((D_m * D_m) / (d * l)) * (h * (((M_m * M_m) * -0.125) / 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 d <= 5.4e-49:
		tmp = w0 * (1.0 + (((D_m * D_m) / (d * l)) * (h * (((M_m * M_m) * -0.125) / 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 (d <= 5.4e-49)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D_m * D_m) / Float64(d * l)) * Float64(h * Float64(Float64(Float64(M_m * M_m) * -0.125) / 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 (d <= 5.4e-49)
		tmp = w0 * (1.0 + (((D_m * D_m) / (d * l)) * (h * (((M_m * M_m) * -0.125) / 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[d, 5.4e-49], N[(w0 * N[(1.0 + N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 5.3999999999999999e-49

   1. Initial program 77.1%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 55.3%

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

   7. Simplified 55.3%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 61.6%

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

   9. Applied egg-rr 61.6%

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

   if 5.3999999999999999e-49 < d

   1. Initial program 86.9%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 84.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{d \cdot \ell} \cdot \left(h \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-96}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot -0.125\right) \cdot \left(M\_m \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right)\\ \end{array} \end{array} \]

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 4e-96)
   w0
   (*
    w0
    (+ 1.0 (* (* D_m D_m) (* (* M_m -0.125) (* M_m (/ h (* 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 <= 4e-96) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D_m * D_m) * ((M_m * -0.125) * (M_m * (h / (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 <= 4d-96) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((d_m * d_m) * ((m_m * (-0.125d0)) * (m_m * (h / (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 <= 4e-96) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D_m * D_m) * ((M_m * -0.125) * (M_m * (h / (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 <= 4e-96:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((D_m * D_m) * ((M_m * -0.125) * (M_m * (h / (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 <= 4e-96)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D_m * D_m) * Float64(Float64(M_m * -0.125) * Float64(M_m * Float64(h / 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 <= 4e-96)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((D_m * D_m) * ((M_m * -0.125) * (M_m * (h / (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, 4e-96], w0, N[(w0 * N[(1.0 + N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * -0.125), $MachinePrecision] * N[(M$95$m * N[(h / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.9999999999999996e-96

   1. Initial program 79.8%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.1%

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

   if 3.9999999999999996e-96 < M

   1. Initial program 81.1%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 50.6%

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

   7. Simplified 50.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 74.2%

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

   9. Applied egg-rr 74.2%

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

Alternative 10: 72.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.7 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d} \cdot \left(w0 \cdot \left(\frac{D\_m}{d} \cdot \left(h \cdot \frac{D\_m}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

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 4.7e+50)
   w0
   (* (/ (* (* M_m M_m) -0.125) d) (* w0 (* (/ D_m d) (* h (/ D_m l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.7e+50) {
		tmp = w0;
	} else {
		tmp = (((M_m * M_m) * -0.125) / d) * (w0 * ((D_m / d) * (h * (D_m / 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 <= 4.7d+50) then
        tmp = w0
    else
        tmp = (((m_m * m_m) * (-0.125d0)) / d) * (w0 * ((d_m / d) * (h * (d_m / 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 <= 4.7e+50) {
		tmp = w0;
	} else {
		tmp = (((M_m * M_m) * -0.125) / d) * (w0 * ((D_m / d) * (h * (D_m / 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 <= 4.7e+50:
		tmp = w0
	else:
		tmp = (((M_m * M_m) * -0.125) / d) * (w0 * ((D_m / d) * (h * (D_m / 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 <= 4.7e+50)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(Float64(M_m * M_m) * -0.125) / d) * Float64(w0 * Float64(Float64(D_m / d) * Float64(h * Float64(D_m / 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 <= 4.7e+50)
		tmp = w0;
	else
		tmp = (((M_m * M_m) * -0.125) / d) * (w0 * ((D_m / d) * (h * (D_m / 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, 4.7e+50], w0, N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(w0 * N[(N[(D$95$m / d), $MachinePrecision] * N[(h * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.69999999999999974e50

   1. Initial program 80.9%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 72.5%

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

   if 4.69999999999999974e50 < M

   1. Initial program 77.6%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 18.0%

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

   10. Simplified 18.0%

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

       1. associate-*l* N/A

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

       2. associate-/r* N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. associate-*l/ N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

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

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

   12. Applied egg-rr 22.6%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.7 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M \cdot M\right) \cdot -0.125}{d} \cdot \left(w0 \cdot \left(\frac{D}{d} \cdot \left(h \cdot \frac{D}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.7 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\frac{D\_m}{d} \cdot \left(\frac{D\_m}{\ell} \cdot \left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot \frac{M\_m}{d}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 5.7e+50)
   w0
   (* w0 (* (/ D_m d) (* (/ D_m l) (* -0.125 (* h (* M_m (/ M_m d)))))))))

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 <= 5.7e+50) {
		tmp = w0;
	} else {
		tmp = w0 * ((D_m / d) * ((D_m / l) * (-0.125 * (h * (M_m * (M_m / d))))));
	}
	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 <= 5.7d+50) then
        tmp = w0
    else
        tmp = w0 * ((d_m / d) * ((d_m / l) * ((-0.125d0) * (h * (m_m * (m_m / d))))))
    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 <= 5.7e+50) {
		tmp = w0;
	} else {
		tmp = w0 * ((D_m / d) * ((D_m / l) * (-0.125 * (h * (M_m * (M_m / d))))));
	}
	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 <= 5.7e+50:
		tmp = w0
	else:
		tmp = w0 * ((D_m / d) * ((D_m / l) * (-0.125 * (h * (M_m * (M_m / d))))))
	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 <= 5.7e+50)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(D_m / d) * Float64(Float64(D_m / l) * Float64(-0.125 * Float64(h * Float64(M_m * Float64(M_m / d)))))));
	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 <= 5.7e+50)
		tmp = w0;
	else
		tmp = w0 * ((D_m / d) * ((D_m / l) * (-0.125 * (h * (M_m * (M_m / d))))));
	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, 5.7e+50], w0, N[(w0 * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * N[(-0.125 * N[(h * N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.7000000000000002e50

   1. Initial program 80.9%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 72.5%

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

   if 5.7000000000000002e50 < M

   1. Initial program 77.6%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r* N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      17. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   10. Simplified 18.0%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d}}{\color{blue}{d \cdot \ell}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{d \cdot \ell}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{d}}{d \cdot \ell}\right)\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d} \cdot h}{\color{blue}{d} \cdot \ell}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}}{\color{blue}{d} \cdot \ell}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{D \cdot D}{d \cdot \ell} \cdot \color{blue}{\left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}\right)\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(\frac{D}{d} \cdot \frac{D}{\ell}\right) \cdot \left(\color{blue}{h} \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{D}{d} \cdot \color{blue}{\left(\frac{D}{\ell} \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{D}{d}\right), \color{blue}{\left(\frac{D}{\ell} \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \left(\color{blue}{\frac{D}{\ell}} \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\left(\frac{D}{\ell}\right), \color{blue}{\left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \left(\color{blue}{h} \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d} \cdot \color{blue}{h}\right)\right)\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \left(\left(\frac{-1}{8} \cdot \frac{M \cdot M}{d}\right) \cdot h\right)\right)\right)\right) \]

       17. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot h\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{M \cdot M}{d} \cdot h\right)}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, d\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \ell\right), \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\frac{M \cdot M}{d}\right), \color{blue}{h}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 23.2%

       \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{\ell} \cdot \left(-0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot h\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.7 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{\ell} \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \left(\left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot \frac{M\_m}{d}\right)\right)\right) \cdot \frac{D\_m}{d \cdot \ell}\right)\right)\\ \end{array} \end{array} \]

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 3.5e+45)
   w0
   (* w0 (* D_m (* (* -0.125 (* h (* M_m (/ M_m d)))) (/ 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 <= 3.5e+45) {
		tmp = w0;
	} else {
		tmp = w0 * (D_m * ((-0.125 * (h * (M_m * (M_m / d)))) * (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 <= 3.5d+45) then
        tmp = w0
    else
        tmp = w0 * (d_m * (((-0.125d0) * (h * (m_m * (m_m / d)))) * (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 <= 3.5e+45) {
		tmp = w0;
	} else {
		tmp = w0 * (D_m * ((-0.125 * (h * (M_m * (M_m / d)))) * (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 <= 3.5e+45:
		tmp = w0
	else:
		tmp = w0 * (D_m * ((-0.125 * (h * (M_m * (M_m / d)))) * (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 <= 3.5e+45)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(D_m * Float64(Float64(-0.125 * Float64(h * Float64(M_m * Float64(M_m / d)))) * Float64(D_m / 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 <= 3.5e+45)
		tmp = w0;
	else
		tmp = w0 * (D_m * ((-0.125 * (h * (M_m * (M_m / d)))) * (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, 3.5e+45], w0, N[(w0 * N[(D$95$m * N[(N[(-0.125 * N[(h * N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.50000000000000023e45

   1. Initial program 80.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 72.5%

      \[\leadsto \color{blue}{w0} \]

   if 3.50000000000000023e45 < M

   1. Initial program 77.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 62.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(d \cdot \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 35.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 35.0%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left(D \cdot D\right) \cdot \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{d \cdot \left(d \cdot \ell\right)}\right)} \]

   8. Taylor expanded in D around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      17. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   10. Simplified 18.0%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d}}{\color{blue}{d \cdot \ell}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{d \cdot \ell}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{d}}{d \cdot \ell}\right)\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d} \cdot h}{\color{blue}{d} \cdot \ell}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot D\right) \cdot \frac{h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}}{\color{blue}{d} \cdot \ell}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{D \cdot D}{d \cdot \ell} \cdot \color{blue}{\left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}\right)\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(D \cdot \frac{D}{d \cdot \ell}\right) \cdot \left(\color{blue}{h} \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \left(D \cdot \color{blue}{\left(\frac{D}{d \cdot \ell} \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{D}{d \cdot \ell} \cdot \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{D}{d \cdot \ell}\right), \color{blue}{\left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \left(d \cdot \ell\right)\right), \left(\color{blue}{h} \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \left(\ell \cdot d\right)\right), \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\ell, d\right)\right), \left(h \cdot \frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\ell, d\right)\right), \left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{d} \cdot \color{blue}{h}\right)\right)\right)\right) \]

       17. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\ell, d\right)\right), \left(\left(\frac{-1}{8} \cdot \frac{M \cdot M}{d}\right) \cdot h\right)\right)\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\ell, d\right)\right), \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot h\right)}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(\ell, d\right)\right), \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{M \cdot M}{d} \cdot h\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 20.9%

       \[\leadsto w0 \cdot \color{blue}{\left(D \cdot \left(\frac{D}{\ell \cdot d} \cdot \left(-0.125 \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot h\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D \cdot \left(\left(-0.125 \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \frac{D}{d \cdot \ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(w0 \cdot \frac{D\_m}{\frac{\ell \cdot \left(d \cdot d\right)}{M\_m \cdot \left(M\_m \cdot \left(h \cdot -0.125\right)\right)}}\right)\\ \end{array} \end{array} \]

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 5.6e+45)
   w0
   (* D_m (* w0 (/ D_m (/ (* l (* d d)) (* M_m (* M_m (* h -0.125)))))))))

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 <= 5.6e+45) {
		tmp = w0;
	} else {
		tmp = D_m * (w0 * (D_m / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	}
	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 <= 5.6d+45) then
        tmp = w0
    else
        tmp = d_m * (w0 * (d_m / ((l * (d * d)) / (m_m * (m_m * (h * (-0.125d0)))))))
    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 <= 5.6e+45) {
		tmp = w0;
	} else {
		tmp = D_m * (w0 * (D_m / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	}
	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 <= 5.6e+45:
		tmp = w0
	else:
		tmp = D_m * (w0 * (D_m / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))))
	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 <= 5.6e+45)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(w0 * Float64(D_m / Float64(Float64(l * Float64(d * d)) / Float64(M_m * Float64(M_m * Float64(h * -0.125)))))));
	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 <= 5.6e+45)
		tmp = w0;
	else
		tmp = D_m * (w0 * (D_m / ((l * (d * d)) / (M_m * (M_m * (h * -0.125))))));
	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, 5.6e+45], w0, N[(D$95$m * N[(w0 * N[(D$95$m / N[(N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(M$95$m * N[(M$95$m * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.5999999999999999e45

   1. Initial program 80.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 72.5%

      \[\leadsto \color{blue}{w0} \]

   if 5.5999999999999999e45 < M

   1. Initial program 77.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 62.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. 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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(d \cdot \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 35.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 35.0%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left(D \cdot D\right) \cdot \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{d \cdot \left(d \cdot \ell\right)}\right)} \]

   8. Taylor expanded in D around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      17. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   10. Simplified 18.0%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{w0} \]

       2. associate-*l* N/A

          \[\leadsto \left(D \cdot \left(D \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot w0 \]

       3. associate-*l* N/A

          \[\leadsto D \cdot \color{blue}{\left(\left(D \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\left(D \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(D \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{w0}\right)\right) \]

   12. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{D \cdot \left(\frac{D}{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}} \cdot w0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(w0 \cdot \frac{D}{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.0% 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 80.2%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 69.8%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))))