Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 87.9%

Time: 19.3s

Alternatives: 17

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{-4}{\frac{D\_m \cdot M\_m}{d\_m}}\\ \mathbf{if}\;d\_m \leq 2 \cdot 10^{-41}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D\_m \cdot M\_m\right)}{d\_m}}{\ell \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M\_m \cdot h}{t\_0} \cdot \frac{D\_m}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ -4.0 (/ (* D_m M_m) d_m))))
   (if (<= d_m 2e-41)
     (* w0 (sqrt (+ 1.0 (/ (/ (* h (* D_m M_m)) d_m) (* l t_0)))))
     (* w0 (sqrt (+ 1.0 (/ (* (/ (* M_m h) t_0) (/ D_m d_m)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = -4.0 / ((D_m * M_m) / d_m);
	double tmp;
	if (d_m <= 2e-41) {
		tmp = w0 * sqrt((1.0 + (((h * (D_m * M_m)) / d_m) / (l * t_0))));
	} else {
		tmp = w0 * sqrt((1.0 + ((((M_m * h) / t_0) * (D_m / d_m)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) / ((d_m * m_m) / d_m_1)
    if (d_m_1 <= 2d-41) then
        tmp = w0 * sqrt((1.0d0 + (((h * (d_m * m_m)) / d_m_1) / (l * t_0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((((m_m * h) / t_0) * (d_m / d_m_1)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = -4.0 / ((D_m * M_m) / d_m);
	double tmp;
	if (d_m <= 2e-41) {
		tmp = w0 * Math.sqrt((1.0 + (((h * (D_m * M_m)) / d_m) / (l * t_0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((M_m * h) / t_0) * (D_m / d_m)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = -4.0 / ((D_m * M_m) / d_m)
	tmp = 0
	if d_m <= 2e-41:
		tmp = w0 * math.sqrt((1.0 + (((h * (D_m * M_m)) / d_m) / (l * t_0))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((((M_m * h) / t_0) * (D_m / d_m)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(-4.0 / Float64(Float64(D_m * M_m) / d_m))
	tmp = 0.0
	if (d_m <= 2e-41)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(D_m * M_m)) / d_m) / Float64(l * t_0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(M_m * h) / t_0) * Float64(D_m / d_m)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = -4.0 / ((D_m * M_m) / d_m);
	tmp = 0.0;
	if (d_m <= 2e-41)
		tmp = w0 * sqrt((1.0 + (((h * (D_m * M_m)) / d_m) / (l * t_0))));
	else
		tmp = w0 * sqrt((1.0 + ((((M_m * h) / t_0) * (D_m / d_m)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(-4.0 / N[(N[(D$95$m * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d$95$m, 2e-41], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(M$95$m * h), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 2.00000000000000001e-41

   1. Initial program 76.9%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 74.9%

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. *-lowering-*.f64 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/l/ N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 78.9%

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

   8. Applied egg-rr 78.9%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 84.2%

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

   10. Applied egg-rr 84.2%

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

   if 2.00000000000000001e-41 < d

   1. Initial program 83.1%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 82.2%

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. *-lowering-*.f64 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/l/ N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.2%

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

   8. Applied egg-rr 84.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 86.7%

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

   10. Applied egg-rr 86.7%

       \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{M \cdot h}{\frac{-4}{\frac{D \cdot M}{d}}} \cdot \frac{D}{d}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-41}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot M\right)}{d}}{\ell \cdot \frac{-4}{\frac{D \cdot M}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M \cdot h}{\frac{-4}{\frac{D \cdot M}{d}}} \cdot \frac{D}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := d\_m \cdot \left(d\_m \cdot \ell\right)\\ \mathbf{if}\;D\_m \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D\_m \cdot D\_m}{d\_m} \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m}\right)\right)\\ \mathbf{elif}\;D\_m \leq 1.8 \cdot 10^{+197}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M\_m \cdot \frac{h}{t\_0}}{\frac{-4}{D\_m \cdot \left(D\_m \cdot M\_m\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 + D\_m \cdot \frac{D\_m}{\frac{t\_0}{-0.125 \cdot \left(h \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* d_m (* d_m l))))
   (if (<= D_m 1.4e+70)
     (*
      w0
      (+
       1.0
       (* (/ -0.125 l) (* (/ (* D_m D_m) d_m) (/ (* h (* M_m M_m)) d_m)))))
     (if (<= D_m 1.8e+197)
       (* w0 (sqrt (+ 1.0 (/ (* M_m (/ h t_0)) (/ -4.0 (* D_m (* D_m M_m)))))))
       (+ w0 (* D_m (/ D_m (/ t_0 (* -0.125 (* h (* M_m (* w0 M_m))))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = d_m * (d_m * l);
	double tmp;
	if (D_m <= 1.4e+70) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	} else if (D_m <= 1.8e+197) {
		tmp = w0 * sqrt((1.0 + ((M_m * (h / t_0)) / (-4.0 / (D_m * (D_m * M_m))))));
	} else {
		tmp = w0 + (D_m * (D_m / (t_0 / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m_1 * (d_m_1 * l)
    if (d_m <= 1.4d+70) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (((d_m * d_m) / d_m_1) * ((h * (m_m * m_m)) / d_m_1))))
    else if (d_m <= 1.8d+197) then
        tmp = w0 * sqrt((1.0d0 + ((m_m * (h / t_0)) / ((-4.0d0) / (d_m * (d_m * m_m))))))
    else
        tmp = w0 + (d_m * (d_m / (t_0 / ((-0.125d0) * (h * (m_m * (w0 * m_m)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = d_m * (d_m * l);
	double tmp;
	if (D_m <= 1.4e+70) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	} else if (D_m <= 1.8e+197) {
		tmp = w0 * Math.sqrt((1.0 + ((M_m * (h / t_0)) / (-4.0 / (D_m * (D_m * M_m))))));
	} else {
		tmp = w0 + (D_m * (D_m / (t_0 / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = d_m * (d_m * l)
	tmp = 0
	if D_m <= 1.4e+70:
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))))
	elif D_m <= 1.8e+197:
		tmp = w0 * math.sqrt((1.0 + ((M_m * (h / t_0)) / (-4.0 / (D_m * (D_m * M_m))))))
	else:
		tmp = w0 + (D_m * (D_m / (t_0 / (-0.125 * (h * (M_m * (w0 * M_m)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(d_m * Float64(d_m * l))
	tmp = 0.0
	if (D_m <= 1.4e+70)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(D_m * D_m) / d_m) * Float64(Float64(h * Float64(M_m * M_m)) / d_m)))));
	elseif (D_m <= 1.8e+197)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M_m * Float64(h / t_0)) / Float64(-4.0 / Float64(D_m * Float64(D_m * M_m)))))));
	else
		tmp = Float64(w0 + Float64(D_m * Float64(D_m / Float64(t_0 / Float64(-0.125 * Float64(h * Float64(M_m * Float64(w0 * M_m))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = d_m * (d_m * l);
	tmp = 0.0;
	if (D_m <= 1.4e+70)
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	elseif (D_m <= 1.8e+197)
		tmp = w0 * sqrt((1.0 + ((M_m * (h / t_0)) / (-4.0 / (D_m * (D_m * M_m))))));
	else
		tmp = w0 + (D_m * (D_m / (t_0 / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 1.4e+70], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D$95$m, 1.8e+197], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M$95$m * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-4.0 / N[(D$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 + N[(D$95$m * N[(D$95$m / N[(t$95$0 / N[(-0.125 * N[(h * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if D < 1.39999999999999995e70

   1. Initial program 79.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in D 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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.4%

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

   7. Simplified 61.4%

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

   if 1.39999999999999995e70 < D < 1.79999999999999991e197

   1. Initial program 85.9%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 82.8%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r/ N/A

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

      7. times-frac N/A

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

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

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

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

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

      10. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. associate-/l* N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. associate-/l/ N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 82.4%

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

   8. Applied egg-rr 82.4%

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

   if 1.79999999999999991e197 < D

   1. Initial program 62.5%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 51.6%

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

   5. Taylor expanded in h around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

   7. Simplified 28.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   9. Applied egg-rr 62.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{+197}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}}{\frac{-4}{D \cdot \left(D \cdot M\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 + D \cdot \frac{D}{\frac{d \cdot \left(d \cdot \ell\right)}{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M\_m \cdot h}{\frac{-4}{\frac{D\_m \cdot M\_m}{d\_m}}} \cdot \frac{D\_m}{d\_m}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -2e-323)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (/ (* M_m h) (/ -4.0 (/ (* D_m M_m) d_m))) (/ D_m d_m)) l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * sqrt((1.0 + ((((M_m * h) / (-4.0 / ((D_m * M_m) / d_m))) * (D_m / d_m)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-2d-323)) then
        tmp = w0 * sqrt((1.0d0 + ((((m_m * h) / ((-4.0d0) / ((d_m * m_m) / d_m_1))) * (d_m / d_m_1)) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * Math.sqrt((1.0 + ((((M_m * h) / (-4.0 / ((D_m * M_m) / d_m))) * (D_m / d_m)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -2e-323:
		tmp = w0 * math.sqrt((1.0 + ((((M_m * h) / (-4.0 / ((D_m * M_m) / d_m))) * (D_m / d_m)) / l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -2e-323)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(M_m * h) / Float64(-4.0 / Float64(Float64(D_m * M_m) / d_m))) * Float64(D_m / d_m)) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -2e-323)
		tmp = w0 * sqrt((1.0 + ((((M_m * h) / (-4.0 / ((D_m * M_m) / d_m))) * (D_m / d_m)) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-323], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(M$95$m * h), $MachinePrecision] / N[(-4.0 / N[(N[(D$95$m * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.7%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. *-lowering-*.f64 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-/l/ N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 79.0%

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

   8. Applied egg-rr 79.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 79.8%

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

   10. Applied egg-rr 79.8%

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

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

   1. Initial program 78.0%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 80.3%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 91.7%

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

Alternative 4: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{-4 \cdot \frac{\frac{d\_m}{D\_m \cdot M\_m}}{D\_m}} \cdot \frac{M\_m}{d\_m}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -2e-323)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (/ h (* -4.0 (/ (/ d_m (* D_m M_m)) D_m))) (/ M_m d_m)) l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 * ((d_m / (D_m * M_m)) / D_m))) * (M_m / d_m)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-2d-323)) then
        tmp = w0 * sqrt((1.0d0 + (((h / ((-4.0d0) * ((d_m_1 / (d_m * m_m)) / d_m))) * (m_m / d_m_1)) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -2e-323) {
		tmp = w0 * Math.sqrt((1.0 + (((h / (-4.0 * ((d_m / (D_m * M_m)) / D_m))) * (M_m / d_m)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -2e-323:
		tmp = w0 * math.sqrt((1.0 + (((h / (-4.0 * ((d_m / (D_m * M_m)) / D_m))) * (M_m / d_m)) / l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -2e-323)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / Float64(-4.0 * Float64(Float64(d_m / Float64(D_m * M_m)) / D_m))) * Float64(M_m / d_m)) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -2e-323)
		tmp = w0 * sqrt((1.0 + (((h / (-4.0 * ((d_m / (D_m * M_m)) / D_m))) * (M_m / d_m)) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-323], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / N[(-4.0 * N[(N[(d$95$m / N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.7%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 75.4%

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

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

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

   6. Applied egg-rr 78.5%

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

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

   1. Initial program 78.0%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 80.3%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 91.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{-4 \cdot \frac{\frac{d}{D \cdot M}}{D}} \cdot \frac{M}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \left(M\_m \cdot \frac{\frac{\frac{D\_m \cdot \left(D\_m \cdot M\_m\right)}{d\_m}}{d\_m}}{-4}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -5e-189)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* h (* M_m (/ (/ (/ (* D_m (* D_m M_m)) d_m) d_m) -4.0))) l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-189) {
		tmp = w0 * sqrt((1.0 + ((h * (M_m * ((((D_m * (D_m * M_m)) / d_m) / d_m) / -4.0))) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-5d-189)) then
        tmp = w0 * sqrt((1.0d0 + ((h * (m_m * ((((d_m * (d_m * m_m)) / d_m_1) / d_m_1) / (-4.0d0)))) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-189) {
		tmp = w0 * Math.sqrt((1.0 + ((h * (M_m * ((((D_m * (D_m * M_m)) / d_m) / d_m) / -4.0))) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -5e-189:
		tmp = w0 * math.sqrt((1.0 + ((h * (M_m * ((((D_m * (D_m * M_m)) / d_m) / d_m) / -4.0))) / l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -5e-189)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(M_m * Float64(Float64(Float64(Float64(D_m * Float64(D_m * M_m)) / d_m) / d_m) / -4.0))) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -5e-189)
		tmp = w0 * sqrt((1.0 + ((h * (M_m * ((((D_m * (D_m * M_m)) / d_m) / d_m) / -4.0))) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-189], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(M$95$m * N[(N[(N[(N[(D$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.9999999999999997e-189

   1. Initial program 80.9%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 77.5%

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

   if -4.9999999999999997e-189 < (/.f64 h l)

   1. Initial program 77.0%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 77.3%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 87.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(D \cdot M\right)}{d}}{d}}{-4}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 1.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m (/ 2.0 M_m)) d_m)))
   (* w0 (sqrt (+ 1.0 (* (/ t_0 l) (/ t_0 (/ -1.0 h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / (2.0 / M_m)) / d_m;
	return w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = (d_m / (2.0d0 / m_m)) / d_m_1
    code = w0 * sqrt((1.0d0 + ((t_0 / l) * (t_0 / ((-1.0d0) / h)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / (2.0 / M_m)) / d_m;
	return w0 * Math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (D_m / (2.0 / M_m)) / d_m
	return w0 * math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m / Float64(2.0 / M_m)) / d_m)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(t_0 / Float64(-1.0 / h))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (D_m / (2.0 / M_m)) / d_m;
	tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. unpow2 N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

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

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

   8. associate-/r* N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

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

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

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

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

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

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

4. Applied egg-rr 87.3%

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

5. Final simplification 87.3%

   \[\leadsto w0 \cdot \sqrt{1 + \frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\ell} \cdot \frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\frac{-1}{h}}} \]
6. Add Preprocessing

Alternative 7: 74.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 2 \cdot 10^{-163}:\\ \;\;\;\;w0 + \left(D\_m \cdot D\_m\right) \cdot \frac{\frac{-0.125 \cdot \left(h \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right)}{d\_m}}{d\_m \cdot \ell}\\ \mathbf{elif}\;d\_m \leq 1.26 \cdot 10^{-42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{h \cdot \left(M\_m \cdot \left(M\_m \cdot \left(D\_m \cdot D\_m\right)\right)\right)}{d\_m \cdot d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 2e-163)
   (+
    w0
    (* (* D_m D_m) (/ (/ (* -0.125 (* h (* M_m (* w0 M_m)))) d_m) (* d_m l))))
   (if (<= d_m 1.26e-42)
     (*
      w0
      (+
       1.0
       (* (/ -0.125 l) (/ (* h (* M_m (* M_m (* D_m D_m)))) (* d_m d_m)))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 2e-163) {
		tmp = w0 + ((D_m * D_m) * (((-0.125 * (h * (M_m * (w0 * M_m)))) / d_m) / (d_m * l)));
	} else if (d_m <= 1.26e-42) {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 2d-163) then
        tmp = w0 + ((d_m * d_m) * ((((-0.125d0) * (h * (m_m * (w0 * m_m)))) / d_m_1) / (d_m_1 * l)))
    else if (d_m_1 <= 1.26d-42) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * ((h * (m_m * (m_m * (d_m * d_m)))) / (d_m_1 * d_m_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 2e-163) {
		tmp = w0 + ((D_m * D_m) * (((-0.125 * (h * (M_m * (w0 * M_m)))) / d_m) / (d_m * l)));
	} else if (d_m <= 1.26e-42) {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 2e-163:
		tmp = w0 + ((D_m * D_m) * (((-0.125 * (h * (M_m * (w0 * M_m)))) / d_m) / (d_m * l)))
	elif d_m <= 1.26e-42:
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 2e-163)
		tmp = Float64(w0 + Float64(Float64(D_m * D_m) * Float64(Float64(Float64(-0.125 * Float64(h * Float64(M_m * Float64(w0 * M_m)))) / d_m) / Float64(d_m * l))));
	elseif (d_m <= 1.26e-42)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(h * Float64(M_m * Float64(M_m * Float64(D_m * D_m)))) / Float64(d_m * d_m)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 2e-163)
		tmp = w0 + ((D_m * D_m) * (((-0.125 * (h * (M_m * (w0 * M_m)))) / d_m) / (d_m * l)));
	elseif (d_m <= 1.26e-42)
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 2e-163], N[(w0 + N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(h * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d$95$m, 1.26e-42], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(M$95$m * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 1.99999999999999985e-163

   1. Initial program 77.8%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 76.2%

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

   5. Taylor expanded in h around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

   7. Simplified 47.2%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 61.5%

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

   9. Applied egg-rr 61.5%

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

   if 1.99999999999999985e-163 < d < 1.26e-42

   1. Initial program 72.7%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 67.3%

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. *-lowering-*.f64 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-/l/ N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.0%

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

   8. Applied egg-rr 84.0%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 89.1%

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

   10. Applied egg-rr 89.1%

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

   11. Taylor expanded in D around 0

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

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

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

       8. associate-*r* N/A

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

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

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

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

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 68.3%

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

   13. Simplified 68.3%

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

   if 1.26e-42 < d

   1. Initial program 82.3%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 81.5%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 78.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-163}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \frac{\frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{d}}{d \cdot \ell}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{-42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{h \cdot \left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 2.35 \cdot 10^{-163}:\\ \;\;\;\;w0 + \left(D\_m \cdot D\_m\right) \cdot \left(\frac{h \cdot -0.125}{d\_m} \cdot \frac{M\_m \cdot \left(w0 \cdot M\_m\right)}{d\_m \cdot \ell}\right)\\ \mathbf{elif}\;d\_m \leq 1.25 \cdot 10^{-42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{h \cdot \left(M\_m \cdot \left(M\_m \cdot \left(D\_m \cdot D\_m\right)\right)\right)}{d\_m \cdot d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 2.35e-163)
   (+
    w0
    (* (* D_m D_m) (* (/ (* h -0.125) d_m) (/ (* M_m (* w0 M_m)) (* d_m l)))))
   (if (<= d_m 1.25e-42)
     (*
      w0
      (+
       1.0
       (* (/ -0.125 l) (/ (* h (* M_m (* M_m (* D_m D_m)))) (* d_m d_m)))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 2.35e-163) {
		tmp = w0 + ((D_m * D_m) * (((h * -0.125) / d_m) * ((M_m * (w0 * M_m)) / (d_m * l))));
	} else if (d_m <= 1.25e-42) {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 2.35d-163) then
        tmp = w0 + ((d_m * d_m) * (((h * (-0.125d0)) / d_m_1) * ((m_m * (w0 * m_m)) / (d_m_1 * l))))
    else if (d_m_1 <= 1.25d-42) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * ((h * (m_m * (m_m * (d_m * d_m)))) / (d_m_1 * d_m_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 2.35e-163) {
		tmp = w0 + ((D_m * D_m) * (((h * -0.125) / d_m) * ((M_m * (w0 * M_m)) / (d_m * l))));
	} else if (d_m <= 1.25e-42) {
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 2.35e-163:
		tmp = w0 + ((D_m * D_m) * (((h * -0.125) / d_m) * ((M_m * (w0 * M_m)) / (d_m * l))))
	elif d_m <= 1.25e-42:
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 2.35e-163)
		tmp = Float64(w0 + Float64(Float64(D_m * D_m) * Float64(Float64(Float64(h * -0.125) / d_m) * Float64(Float64(M_m * Float64(w0 * M_m)) / Float64(d_m * l)))));
	elseif (d_m <= 1.25e-42)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(h * Float64(M_m * Float64(M_m * Float64(D_m * D_m)))) / Float64(d_m * d_m)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 2.35e-163)
		tmp = w0 + ((D_m * D_m) * (((h * -0.125) / d_m) * ((M_m * (w0 * M_m)) / (d_m * l))));
	elseif (d_m <= 1.25e-42)
		tmp = w0 * (1.0 + ((-0.125 / l) * ((h * (M_m * (M_m * (D_m * D_m)))) / (d_m * d_m))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 2.35e-163], N[(w0 + N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(h * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d$95$m, 1.25e-42], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(M$95$m * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 2.35e-163

   1. Initial program 77.8%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 76.2%

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

   5. Taylor expanded in h around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

   7. Simplified 47.2%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 58.9%

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

   9. Applied egg-rr 58.9%

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

   if 2.35e-163 < d < 1.25000000000000001e-42

   1. Initial program 72.7%

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

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

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

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

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

      3. sub-neg N/A

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

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

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

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

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

   3. Simplified 67.3%

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. *-lowering-*.f64 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-/l/ N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.0%

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

   8. Applied egg-rr 84.0%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 89.1%

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

   10. Applied egg-rr 89.1%

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

   11. Taylor expanded in D around 0

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

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

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

       8. associate-*r* N/A

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

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

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

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

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 68.3%

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

   13. Simplified 68.3%

       \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \frac{\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h}{d \cdot d}\right)} \]

   if 1.25000000000000001e-42 < d

   1. Initial program 82.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.35 \cdot 10^{-163}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \left(\frac{h \cdot -0.125}{d} \cdot \frac{M \cdot \left(w0 \cdot M\right)}{d \cdot \ell}\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-42}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \frac{h \cdot \left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.35 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D\_m \cdot D\_m}{d\_m} \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + D\_m \cdot \frac{D\_m}{\frac{d\_m \cdot \left(d\_m \cdot \ell\right)}{-0.125 \cdot \left(h \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 1.35e+70)
   (*
    w0
    (+ 1.0 (* (/ -0.125 l) (* (/ (* D_m D_m) d_m) (/ (* h (* M_m M_m)) d_m)))))
   (+
    w0
    (*
     D_m
     (/ D_m (/ (* d_m (* d_m l)) (* -0.125 (* h (* M_m (* w0 M_m))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.35e+70) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	} else {
		tmp = w0 + (D_m * (D_m / ((d_m * (d_m * l)) / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 1.35d+70) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (((d_m * d_m) / d_m_1) * ((h * (m_m * m_m)) / d_m_1))))
    else
        tmp = w0 + (d_m * (d_m / ((d_m_1 * (d_m_1 * l)) / ((-0.125d0) * (h * (m_m * (w0 * m_m)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.35e+70) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	} else {
		tmp = w0 + (D_m * (D_m / ((d_m * (d_m * l)) / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 1.35e+70:
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))))
	else:
		tmp = w0 + (D_m * (D_m / ((d_m * (d_m * l)) / (-0.125 * (h * (M_m * (w0 * M_m)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.35e+70)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(D_m * D_m) / d_m) * Float64(Float64(h * Float64(M_m * M_m)) / d_m)))));
	else
		tmp = Float64(w0 + Float64(D_m * Float64(D_m / Float64(Float64(d_m * Float64(d_m * l)) / Float64(-0.125 * Float64(h * Float64(M_m * Float64(w0 * M_m))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 1.35e+70)
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	else
		tmp = w0 + (D_m * (D_m / ((d_m * (d_m * l)) / (-0.125 * (h * (M_m * (w0 * M_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 1.35e+70], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(D$95$m * N[(D$95$m / N[(N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / N[(-0.125 * N[(h * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.35e70

   1. Initial program 79.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{2}}{d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{D \cdot M}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{M}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{1}{\frac{2}{M}}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{D}{\frac{2}{M}}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \left(\frac{2}{M}\right)\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(2, M\right)\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(2, M\right)\right), d\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 89.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\ell} \cdot \frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\frac{1}{h}}}} \]

   5. Taylor expanded in D 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. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 61.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right)\right)\right)\right)\right) \]

   7. Simplified 61.4%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]

   if 1.35e70 < D

   1. Initial program 76.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 49.6%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(D \cdot \color{blue}{\left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{D}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot \frac{\frac{-1}{8} \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{D}\right)\right) \]

   9. Applied egg-rr 69.7%

      \[\leadsto w0 + \color{blue}{\frac{D}{\frac{d \cdot \left(d \cdot \ell\right)}{-0.125 \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}} \cdot D} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.35 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + D \cdot \frac{D}{\frac{d \cdot \left(d \cdot \ell\right)}{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.7 \cdot 10^{-48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D\_m \cdot D\_m}{d\_m} \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3.7e-48)
   w0
   (*
    w0
    (+
     1.0
     (* (/ -0.125 l) (* (/ (* D_m D_m) d_m) (/ (* h (* M_m M_m)) d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.7e-48) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 3.7d-48) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (((d_m * d_m) / d_m_1) * ((h * (m_m * m_m)) / d_m_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.7e-48) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3.7e-48:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3.7e-48)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(D_m * D_m) / d_m) * Float64(Float64(h * Float64(M_m * M_m)) / d_m)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3.7e-48)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((-0.125 / l) * (((D_m * D_m) / d_m) * ((h * (M_m * M_m)) / d_m))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3.7e-48], w0, N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.6999999999999998e-48

   1. Initial program 80.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 72.7%

      \[\leadsto \color{blue}{w0} \]

   if 3.6999999999999998e-48 < M

   1. Initial program 75.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{2}}{d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{D \cdot M}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{M}{2}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{1}{\frac{2}{M}}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{D}{\frac{2}{M}}\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \left(\frac{2}{M}\right)\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(2, M\right)\right), d\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(2, M\right)\right), d\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 81.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\ell} \cdot \frac{\frac{\frac{D}{\frac{2}{M}}}{d}}{\frac{1}{h}}}} \]

   5. Taylor expanded in D 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. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \color{blue}{d}}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{{D}^{2}}{d} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{d}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\left(\frac{{D}^{2}}{d}\right), \color{blue}{\left(\frac{{M}^{2} \cdot h}{d}\right)}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({D}^{2}\right), d\right), \left(\frac{\color{blue}{{M}^{2} \cdot h}}{d}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \left(\frac{\color{blue}{{M}^{2}} \cdot h}{d}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \color{blue}{d}\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\left(h \cdot {M}^{2}\right), d\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left({M}^{2}\right)\right), d\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot M\right)\right), d\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 57.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right), d\right)\right)\right)\right)\right) \]

   7. Simplified 57.6%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.7 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{D\_m \cdot D\_m}{d\_m} \cdot \left(\left(h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d\_m}\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3.7e+20)
   w0
   (/ (* (/ (* D_m D_m) d_m) (* (* h (* w0 (* M_m M_m))) (/ -0.125 d_m))) l)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.7e+20) {
		tmp = w0;
	} else {
		tmp = (((D_m * D_m) / d_m) * ((h * (w0 * (M_m * M_m))) * (-0.125 / d_m))) / l;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 3.7d+20) then
        tmp = w0
    else
        tmp = (((d_m * d_m) / d_m_1) * ((h * (w0 * (m_m * m_m))) * ((-0.125d0) / d_m_1))) / l
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.7e+20) {
		tmp = w0;
	} else {
		tmp = (((D_m * D_m) / d_m) * ((h * (w0 * (M_m * M_m))) * (-0.125 / d_m))) / l;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3.7e+20:
		tmp = w0
	else:
		tmp = (((D_m * D_m) / d_m) * ((h * (w0 * (M_m * M_m))) * (-0.125 / d_m))) / l
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3.7e+20)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) / d_m) * Float64(Float64(h * Float64(w0 * Float64(M_m * M_m))) * Float64(-0.125 / d_m))) / l);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3.7e+20)
		tmp = w0;
	else
		tmp = (((D_m * D_m) / d_m) * ((h * (w0 * (M_m * M_m))) * (-0.125 / d_m))) / l;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3.7e+20], w0, N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.7e20

   1. Initial program 80.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   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 3.7e20 < M

   1. Initial program 73.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(h \cdot {M}^{2}\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right) \]

      18. *-lowering-*.f64 23.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right) \]

   10. Simplified 23.2%

       \[\leadsto \color{blue}{\frac{\left(\left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{d \cdot d}}{\color{blue}{\ell}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{d \cdot d}\right), \color{blue}{\ell}\right) \]

   12. Applied egg-rr 29.8%

       \[\leadsto \color{blue}{\frac{\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right) \cdot \frac{D \cdot D}{d}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{+20}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{D \cdot D}{d} \cdot \left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.05 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d\_m} \cdot \frac{D\_m}{d\_m \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 1.05e+95)
   w0
   (* (/ (* -0.125 (* D_m (* h (* w0 (* M_m M_m))))) d_m) (/ D_m (* d_m l)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.05e+95) {
		tmp = w0;
	} else {
		tmp = ((-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) / d_m) * (D_m / (d_m * l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 1.05d+95) then
        tmp = w0
    else
        tmp = (((-0.125d0) * (d_m * (h * (w0 * (m_m * m_m))))) / d_m_1) * (d_m / (d_m_1 * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.05e+95) {
		tmp = w0;
	} else {
		tmp = ((-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) / d_m) * (D_m / (d_m * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 1.05e+95:
		tmp = w0
	else:
		tmp = ((-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) / d_m) * (D_m / (d_m * l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 1.05e+95)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64(D_m * Float64(h * Float64(w0 * Float64(M_m * M_m))))) / d_m) * Float64(D_m / Float64(d_m * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 1.05e+95)
		tmp = w0;
	else
		tmp = ((-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) / d_m) * (D_m / (d_m * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 1.05e+95], w0, N[(N[(N[(-0.125 * N[(D$95$m * N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.05e95

   1. Initial program 79.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   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 1.05e95 < M

   1. Initial program 75.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 32.7%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(h \cdot {M}^{2}\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right) \]

      18. *-lowering-*.f64 26.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right) \]

   10. Simplified 26.1%

       \[\leadsto \color{blue}{\frac{\left(\left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D\right) \cdot D}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       2. associate-*l* N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D\right) \cdot D}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       3. times-frac N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D}{d} \cdot \color{blue}{\frac{D}{d \cdot \ell}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D}{d}\right), \color{blue}{\left(\frac{D}{d \cdot \ell}\right)}\right) \]

   12. Applied egg-rr 32.3%

       \[\leadsto \color{blue}{\frac{-0.125 \cdot \left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right) \cdot D\right)}{d} \cdot \frac{D}{d \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(D \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)\right)}{d} \cdot \frac{D}{d \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)\right) \cdot \frac{D\_m}{d\_m \cdot \left(d\_m \cdot \ell\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3.5e+95)
   w0
   (* (* -0.125 (* D_m (* h (* w0 (* M_m M_m))))) (/ D_m (* d_m (* d_m l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.5e+95) {
		tmp = w0;
	} else {
		tmp = (-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) * (D_m / (d_m * (d_m * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 3.5d+95) then
        tmp = w0
    else
        tmp = ((-0.125d0) * (d_m * (h * (w0 * (m_m * m_m))))) * (d_m / (d_m_1 * (d_m_1 * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.5e+95) {
		tmp = w0;
	} else {
		tmp = (-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) * (D_m / (d_m * (d_m * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3.5e+95:
		tmp = w0
	else:
		tmp = (-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) * (D_m / (d_m * (d_m * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3.5e+95)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 * Float64(D_m * Float64(h * Float64(w0 * Float64(M_m * M_m))))) * Float64(D_m / Float64(d_m * Float64(d_m * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3.5e+95)
		tmp = w0;
	else
		tmp = (-0.125 * (D_m * (h * (w0 * (M_m * M_m))))) * (D_m / (d_m * (d_m * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3.5e+95], w0, N[(N[(-0.125 * N[(D$95$m * N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.5e95

   1. Initial program 79.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 72.6%

      \[\leadsto \color{blue}{w0} \]

   if 3.5e95 < M

   1. Initial program 75.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 31.2%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot w0\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot h\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(h \cdot {M}^{2}\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left({M}^{2}\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \left(M \cdot M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right) \]

      18. *-lowering-*.f64 26.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, M\right)\right)\right), w0\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right) \]

   10. Simplified 26.7%

       \[\leadsto \color{blue}{\frac{\left(\left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D\right) \cdot D}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       2. associate-/l* N/A

          \[\leadsto \left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D\right) \cdot \color{blue}{\frac{D}{\left(d \cdot d\right) \cdot \ell}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot w0\right) \cdot D\right), \color{blue}{\left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)}\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w0\right)\right) \cdot D\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot w0\right)\right)\right) \cdot D\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)\right) \cdot D\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left(\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right) \cdot D\right)\right), \left(\frac{\color{blue}{D}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right) \cdot D\right)\right), \left(\frac{\color{blue}{D}}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \left(M \cdot w0\right)\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(\left(M \cdot M\right) \cdot w0\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(w0 \cdot \left(M \cdot M\right)\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), D\right)\right), \left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), D\right)\right), \mathsf{/.f64}\left(D, \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), D\right)\right), \mathsf{/.f64}\left(D, \left(d \cdot \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right) \]

       18. *-lowering-*.f64 32.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), D\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right) \]

   12. Applied egg-rr 32.4%

       \[\leadsto \color{blue}{\left(-0.125 \cdot \left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right) \cdot D\right)\right) \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \left(D \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)\right)\right) \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.42 \cdot 10^{+80}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{w0 \cdot \left(M\_m \cdot M\_m\right)}{M\_m \cdot M\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{w0}{M\_m} \cdot \frac{M\_m \cdot M\_m}{M\_m}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 1.42e+80)
   w0
   (if (<= M_m 1.32e+154)
     (/ (* w0 (* M_m M_m)) (* M_m M_m))
     (* (/ w0 M_m) (/ (* M_m M_m) M_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.42e+80) {
		tmp = w0;
	} else if (M_m <= 1.32e+154) {
		tmp = (w0 * (M_m * M_m)) / (M_m * M_m);
	} else {
		tmp = (w0 / M_m) * ((M_m * M_m) / M_m);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 1.42d+80) then
        tmp = w0
    else if (m_m <= 1.32d+154) then
        tmp = (w0 * (m_m * m_m)) / (m_m * m_m)
    else
        tmp = (w0 / m_m) * ((m_m * m_m) / m_m)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.42e+80) {
		tmp = w0;
	} else if (M_m <= 1.32e+154) {
		tmp = (w0 * (M_m * M_m)) / (M_m * M_m);
	} else {
		tmp = (w0 / M_m) * ((M_m * M_m) / M_m);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 1.42e+80:
		tmp = w0
	elif M_m <= 1.32e+154:
		tmp = (w0 * (M_m * M_m)) / (M_m * M_m)
	else:
		tmp = (w0 / M_m) * ((M_m * M_m) / M_m)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 1.42e+80)
		tmp = w0;
	elseif (M_m <= 1.32e+154)
		tmp = Float64(Float64(w0 * Float64(M_m * M_m)) / Float64(M_m * M_m));
	else
		tmp = Float64(Float64(w0 / M_m) * Float64(Float64(M_m * M_m) / M_m));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 1.42e+80)
		tmp = w0;
	elseif (M_m <= 1.32e+154)
		tmp = (w0 * (M_m * M_m)) / (M_m * M_m);
	else
		tmp = (w0 / M_m) * ((M_m * M_m) / M_m);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 1.42e+80], w0, If[LessEqual[M$95$m, 1.32e+154], N[(N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(w0 / M$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.4200000000000001e80

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   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 1.4200000000000001e80 < M < 1.31999999999999998e154

   1. Initial program 67.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 72.8%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 34.4%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({M}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(M \cdot M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left({D}^{2} \cdot h\right) \cdot w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({D}^{2} \cdot h\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(h \cdot {D}^{2}\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left({D}^{2}\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 40.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right) \]

   10. Simplified 40.0%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\frac{-0.125 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell} + \frac{w0}{M \cdot M}\right)} \]

   11. Taylor expanded in h around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right) \]

       3. *-lowering-*.f64 26.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right) \]

   13. Simplified 26.7%

       \[\leadsto \left(M \cdot M\right) \cdot \color{blue}{\frac{w0}{M \cdot M}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot w0}{\color{blue}{M \cdot M}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot w0\right), \color{blue}{\left(M \cdot M\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(w0 \cdot \left(M \cdot M\right)\right), \left(\color{blue}{M} \cdot M\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right), \left(\color{blue}{M} \cdot M\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right), \left(M \cdot M\right)\right) \]

       6. *-lowering-*.f64 55.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right), \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right) \]

   15. Applied egg-rr 55.0%

       \[\leadsto \color{blue}{\frac{w0 \cdot \left(M \cdot M\right)}{M \cdot M}} \]

   if 1.31999999999999998e154 < M

   1. Initial program 77.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 31.5%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({M}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(M \cdot M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left({D}^{2} \cdot h\right) \cdot w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({D}^{2} \cdot h\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(h \cdot {D}^{2}\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left({D}^{2}\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 31.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right) \]

   10. Simplified 31.4%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\frac{-0.125 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell} + \frac{w0}{M \cdot M}\right)} \]

   11. Taylor expanded in h around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right) \]

       3. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right) \]

   13. Simplified 0.0%

       \[\leadsto \left(M \cdot M\right) \cdot \color{blue}{\frac{w0}{M \cdot M}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot w0}{\color{blue}{M \cdot M}} \]

       2. times-frac N/A

          \[\leadsto \frac{M \cdot M}{M} \cdot \color{blue}{\frac{w0}{M}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{M \cdot M}{M}\right), \color{blue}{\left(\frac{w0}{M}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), M\right), \left(\frac{\color{blue}{w0}}{M}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), M\right), \left(\frac{w0}{M}\right)\right) \]

       6. /-lowering-/.f64 36.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), M\right), \mathsf{/.f64}\left(w0, \color{blue}{M}\right)\right) \]

   15. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{\frac{M \cdot M}{M} \cdot \frac{w0}{M}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.42 \cdot 10^{+80}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{w0 \cdot \left(M \cdot M\right)}{M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\frac{w0}{M} \cdot \frac{M \cdot M}{M}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3 \cdot 10^{+78}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{w0 \cdot \left(M\_m \cdot M\_m\right)}{M\_m}}{M\_m}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3e+78) w0 (/ (/ (* w0 (* M_m M_m)) M_m) M_m)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3e+78) {
		tmp = w0;
	} else {
		tmp = ((w0 * (M_m * M_m)) / M_m) / M_m;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 3d+78) then
        tmp = w0
    else
        tmp = ((w0 * (m_m * m_m)) / m_m) / m_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3e+78) {
		tmp = w0;
	} else {
		tmp = ((w0 * (M_m * M_m)) / M_m) / M_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3e+78:
		tmp = w0
	else:
		tmp = ((w0 * (M_m * M_m)) / M_m) / M_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3e+78)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(w0 * Float64(M_m * M_m)) / M_m) / M_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3e+78)
		tmp = w0;
	else
		tmp = ((w0 * (M_m * M_m)) / M_m) / M_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3e+78], w0, N[(N[(N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision] / M$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.99999999999999982e78

   1. Initial program 80.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   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 2.99999999999999982e78 < M

   1. Initial program 72.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 32.8%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({M}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(M \cdot M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left({D}^{2} \cdot h\right) \cdot w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({D}^{2} \cdot h\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(h \cdot {D}^{2}\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left({D}^{2}\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 35.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right) \]

   10. Simplified 35.2%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\frac{-0.125 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell} + \frac{w0}{M \cdot M}\right)} \]

   11. Taylor expanded in h around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right) \]

       3. *-lowering-*.f64 11.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right) \]

   13. Simplified 11.9%

       \[\leadsto \left(M \cdot M\right) \cdot \color{blue}{\frac{w0}{M \cdot M}} \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot w0}{\color{blue}{M \cdot M}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(M \cdot M\right) \cdot w0}{M}}{\color{blue}{M}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot w0}{M}\right), \color{blue}{M}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot w0\right), M\right), M\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(w0 \cdot \left(M \cdot M\right)\right), M\right), M\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right), M\right), M\right) \]

       7. *-lowering-*.f64 47.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right), M\right), M\right) \]

   15. Applied egg-rr 47.5%

       \[\leadsto \color{blue}{\frac{\frac{w0 \cdot \left(M \cdot M\right)}{M}}{M}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 70.3% accurate, 15.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.5 \cdot 10^{-69}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M\_m \cdot M\_m\right) \cdot \frac{w0}{M\_m}}{M\_m}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 7.5e-69) w0 (/ (* (* M_m M_m) (/ w0 M_m)) M_m)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 7.5e-69) {
		tmp = w0;
	} else {
		tmp = ((M_m * M_m) * (w0 / M_m)) / M_m;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 7.5d-69) then
        tmp = w0
    else
        tmp = ((m_m * m_m) * (w0 / m_m)) / m_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 7.5e-69) {
		tmp = w0;
	} else {
		tmp = ((M_m * M_m) * (w0 / M_m)) / M_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 7.5e-69:
		tmp = w0
	else:
		tmp = ((M_m * M_m) * (w0 / M_m)) / M_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 7.5e-69)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(M_m * M_m) * Float64(w0 / M_m)) / M_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 7.5e-69)
		tmp = w0;
	else
		tmp = ((M_m * M_m) * (w0 / M_m)) / M_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 7.5e-69], w0, N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(w0 / M$95$m), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.5e-69

   1. Initial program 80.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 72.1%

      \[\leadsto \color{blue}{w0} \]

   if 7.5e-69 < M

   1. Initial program 75.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

   7. Simplified 44.7%

      \[\leadsto \color{blue}{w0 + \left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({M}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} + \frac{w0}{{M}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(M \cdot M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} + \frac{w0}{{M}^{2}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{\color{blue}{w0}}{{M}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left({D}^{2} \cdot h\right) \cdot w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({D}^{2} \cdot h\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(h \cdot {D}^{2}\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left({D}^{2}\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(D \cdot D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \left({d}^{2} \cdot \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \left(\frac{w0}{{M}^{2}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right)\right) \]

      19. *-lowering-*.f64 41.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(D, D\right)\right), w0\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right)\right) \]

   10. Simplified 41.0%

       \[\leadsto \color{blue}{\left(M \cdot M\right) \cdot \left(\frac{-0.125 \cdot \left(\left(h \cdot \left(D \cdot D\right)\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell} + \frac{w0}{M \cdot M}\right)} \]

   11. Taylor expanded in h around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \color{blue}{\left(\frac{w0}{{M}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \color{blue}{\left({M}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \left(M \cdot \color{blue}{M}\right)\right)\right) \]

       3. *-lowering-*.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{/.f64}\left(w0, \mathsf{*.f64}\left(M, \color{blue}{M}\right)\right)\right) \]

   13. Simplified 34.4%

       \[\leadsto \left(M \cdot M\right) \cdot \color{blue}{\frac{w0}{M \cdot M}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{w0}{M \cdot M} \cdot \color{blue}{\left(M \cdot M\right)} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{w0}{M}}{M} \cdot \left(\color{blue}{M} \cdot M\right) \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{w0}{M} \cdot \left(M \cdot M\right)}{\color{blue}{M}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{w0}{M} \cdot \left(M \cdot M\right)\right), \color{blue}{M}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{w0}{M}\right), \left(M \cdot M\right)\right), M\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w0, M\right), \left(M \cdot M\right)\right), M\right) \]

       7. *-lowering-*.f64 53.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(w0, M\right), \mathsf{*.f64}\left(M, M\right)\right), M\right) \]

   15. Applied egg-rr 53.2%

       \[\leadsto \color{blue}{\frac{\frac{w0}{M} \cdot \left(M \cdot M\right)}{M}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{-69}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M \cdot M\right) \cdot \frac{w0}{M}}{M}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.0% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.0%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)\right)\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}{\ell}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

3. Simplified 77.4%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(M \cdot \frac{\frac{\frac{D \cdot \left(M \cdot D\right)}{d}}{d}}{-4}\right)}{\ell}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 66.5%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))