Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 88.6%

Time: 17.5s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{\frac{d}{M\_m}}\\ w0 \cdot \sqrt{1 + \frac{\frac{t\_0}{\frac{\frac{-4}{h}}{t\_0}}}{\ell}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m / (d / M_m)
	return w0 * math.sqrt((1.0 + ((t_0 / ((-4.0 / h) / t_0)) / l)))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d / M_m))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / Float64(Float64(-4.0 / h) / t_0)) / l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m / (d / M_m);
	tmp = w0 * sqrt((1.0 + ((t_0 / ((-4.0 / h) / t_0)) / l)));
end

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 79.3%

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. times-frac N/A

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

   13. /-lowering-/.f64 82.0%

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

6. Applied egg-rr 82.0%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

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

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

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

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

   5. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   13. associate-/r/ N/A

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

   14. clear-num N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

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

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

   18. /-lowering-/.f64 86.2%

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

8. Applied egg-rr 86.2%

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

   1. associate-/r/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f64 88.5%

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

10. Applied egg-rr 88.5%

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

    1. associate-*r/ N/A

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

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

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

    3. *-commutative N/A

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

    4. clear-num N/A

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

    5. clear-num N/A

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

    6. frac-times N/A

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

    7. metadata-eval N/A

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

    8. un-div-inv N/A

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

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

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

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

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

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

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

    12. clear-num N/A

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

    13. un-div-inv N/A

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

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

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

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

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

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

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

    17. /-lowering-/.f64 89.2%

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

12. Applied egg-rr 89.2%

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

Alternative 2: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{\frac{d}{M\_m}}\\ w0 \cdot \sqrt{1 + \left(t\_0 \cdot \frac{h}{-4}\right) \cdot \frac{t\_0}{\ell}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m / (d / M_m)
	return w0 * math.sqrt((1.0 + ((t_0 * (h / -4.0)) * (t_0 / l))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d / M_m))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 * Float64(h / -4.0)) * Float64(t_0 / l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m / (d / M_m);
	tmp = w0 * sqrt((1.0 + ((t_0 * (h / -4.0)) * (t_0 / l))));
end

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 79.3%

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. times-frac N/A

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

   13. /-lowering-/.f64 82.0%

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

6. Applied egg-rr 82.0%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

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

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

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

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

   5. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   13. associate-/r/ N/A

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

   14. clear-num N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

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

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

   18. /-lowering-/.f64 86.2%

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

8. Applied egg-rr 86.2%

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

   1. associate-/r/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f64 88.5%

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

10. Applied egg-rr 88.5%

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

11. Final simplification 88.5%

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

Alternative 3: 71.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-98}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \frac{\frac{h \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \left(w0 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 2e-98)
   (+ w0 (* (/ -0.125 l) (/ (/ (* h (* (* D_m D_m) (* M_m (* w0 M_m)))) d) d)))
   (+
    w0
    (* (/ -0.125 l) (* (/ D_m d) (* (/ D_m d) (* w0 (* h (* M_m M_m)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2e-98) {
		tmp = w0 + ((-0.125 / l) * (((h * ((D_m * D_m) * (M_m * (w0 * M_m)))) / d) / d));
	} else {
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 2d-98) then
        tmp = w0 + (((-0.125d0) / l) * (((h * ((d_m * d_m) * (m_m * (w0 * m_m)))) / d) / d))
    else
        tmp = w0 + (((-0.125d0) / l) * ((d_m / d) * ((d_m / d) * (w0 * (h * (m_m * m_m))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2e-98) {
		tmp = w0 + ((-0.125 / l) * (((h * ((D_m * D_m) * (M_m * (w0 * M_m)))) / d) / d));
	} else {
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 2e-98:
		tmp = w0 + ((-0.125 / l) * (((h * ((D_m * D_m) * (M_m * (w0 * M_m)))) / d) / d))
	else:
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 2e-98)
		tmp = Float64(w0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(h * Float64(Float64(D_m * D_m) * Float64(M_m * Float64(w0 * M_m)))) / d) / d)));
	else
		tmp = Float64(w0 + Float64(Float64(-0.125 / l) * Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(w0 * Float64(h * Float64(M_m * M_m)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 2e-98)
		tmp = w0 + ((-0.125 / l) * (((h * ((D_m * D_m) * (M_m * (w0 * M_m)))) / d) / d));
	else
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 2e-98], N[(w0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(h * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(w0 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.99999999999999988e-98

   1. Initial program 76.2%

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

   3. Simplified 79.0%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 41.2%

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

   7. Simplified 41.2%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 57.6%

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

   9. Applied egg-rr 57.6%

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

   if 1.99999999999999988e-98 < d

   1. Initial program 83.0%

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

   3. Simplified 80.0%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 62.5%

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

   7. Simplified 62.5%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 69.0%

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

   9. Applied egg-rr 69.0%

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

       1. associate-/l* N/A

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

       2. div-inv N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. div-inv N/A

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

       7. clear-num N/A

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

       8. /-rgt-identity N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

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

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

       18. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-98}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \frac{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-79}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\left(w0 \cdot D\_m\right) \cdot \left(D\_m \cdot \frac{M\_m \cdot h}{\frac{d}{\frac{M\_m}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \left(w0 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 4e-79)
   (+ w0 (* (/ -0.125 l) (* (* w0 D_m) (* D_m (/ (* M_m h) (/ d (/ M_m d)))))))
   (+
    w0
    (* (/ -0.125 l) (* (/ D_m d) (* (/ D_m d) (* w0 (* h (* M_m M_m)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4e-79) {
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	} else {
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 4d-79) then
        tmp = w0 + (((-0.125d0) / l) * ((w0 * d_m) * (d_m * ((m_m * h) / (d / (m_m / d))))))
    else
        tmp = w0 + (((-0.125d0) / l) * ((d_m / d) * ((d_m / d) * (w0 * (h * (m_m * m_m))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4e-79) {
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	} else {
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 4e-79:
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))))
	else:
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 4e-79)
		tmp = Float64(w0 + Float64(Float64(-0.125 / l) * Float64(Float64(w0 * D_m) * Float64(D_m * Float64(Float64(M_m * h) / Float64(d / Float64(M_m / d)))))));
	else
		tmp = Float64(w0 + Float64(Float64(-0.125 / l) * Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(w0 * Float64(h * Float64(M_m * M_m)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 4e-79)
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	else
		tmp = w0 + ((-0.125 / l) * ((D_m / d) * ((D_m / d) * (w0 * (h * (M_m * M_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 4e-79], N[(w0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(w0 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m * h), $MachinePrecision] / N[(d / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(w0 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4e-79

   1. Initial program 76.4%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 41.5%

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

   7. Simplified 41.5%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 58.0%

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

   9. Applied egg-rr 58.0%

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

       1. associate-/l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. clear-num N/A

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

       6. associate-/r* N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. associate-/r* N/A

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

       10. clear-num N/A

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

       11. times-frac N/A

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

       12. associate-/l* N/A

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

       13. clear-num N/A

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

   11. Applied egg-rr 62.4%

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

   if 4e-79 < d

   1. Initial program 82.8%

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

   3. Simplified 79.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 62.1%

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

   7. Simplified 62.1%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 68.7%

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

   9. Applied egg-rr 68.7%

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

       1. associate-/l* N/A

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

       2. div-inv N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. div-inv N/A

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

       7. clear-num N/A

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

       8. /-rgt-identity N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

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

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

       18. *-lowering-*.f64 71.4%

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

   11. Applied egg-rr 71.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-79}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\left(w0 \cdot D\right) \cdot \left(D \cdot \frac{M \cdot h}{\frac{d}{\frac{M}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8.5 \cdot 10^{-28}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\left(w0 \cdot D\_m\right) \cdot \left(D\_m \cdot \frac{M\_m \cdot h}{\frac{d}{\frac{M\_m}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 8.5e-28)
   (+ w0 (* (/ -0.125 l) (* (* w0 D_m) (* D_m (/ (* M_m h) (/ d (/ M_m d)))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 8.5e-28) {
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 8.5d-28) then
        tmp = w0 + (((-0.125d0) / l) * ((w0 * d_m) * (d_m * ((m_m * h) / (d / (m_m / d))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 8.5e-28) {
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 8.5e-28:
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 8.5e-28)
		tmp = Float64(w0 + Float64(Float64(-0.125 / l) * Float64(Float64(w0 * D_m) * Float64(D_m * Float64(Float64(M_m * h) / Float64(d / Float64(M_m / d)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 8.5e-28)
		tmp = w0 + ((-0.125 / l) * ((w0 * D_m) * (D_m * ((M_m * h) / (d / (M_m / d))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 8.5e-28], N[(w0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(w0 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m * h), $MachinePrecision] / N[(d / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 8.49999999999999925e-28

   1. Initial program 76.0%

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

   3. Simplified 78.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 41.4%

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

   7. Simplified 41.4%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 57.6%

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

   9. Applied egg-rr 57.6%

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

       1. associate-/l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. clear-num N/A

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

       6. associate-/r* N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. associate-/r* N/A

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

       10. clear-num N/A

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

       11. times-frac N/A

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

       12. associate-/l* N/A

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

       13. clear-num N/A

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

   11. Applied egg-rr 62.4%

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

   if 8.49999999999999925e-28 < d

   1. Initial program 84.3%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 82.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.5 \cdot 10^{-28}:\\ \;\;\;\;w0 + \frac{-0.125}{\ell} \cdot \left(\left(w0 \cdot D\right) \cdot \left(D \cdot \frac{M \cdot h}{\frac{d}{\frac{M}{d}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{M\_m \cdot \frac{M\_m \cdot \left(w0 \cdot \left(D\_m \cdot h\right)\right)}{\frac{d}{\frac{D\_m}{-8}}}}{d \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2.3e+49)
   w0
   (/ (* M_m (/ (* M_m (* w0 (* D_m h))) (/ d (/ D_m -8.0)))) (* d l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.3e+49) {
		tmp = w0;
	} else {
		tmp = (M_m * ((M_m * (w0 * (D_m * h))) / (d / (D_m / -8.0)))) / (d * l);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2.3d+49) then
        tmp = w0
    else
        tmp = (m_m * ((m_m * (w0 * (d_m * h))) / (d / (d_m / (-8.0d0))))) / (d * l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.3e+49) {
		tmp = w0;
	} else {
		tmp = (M_m * ((M_m * (w0 * (D_m * h))) / (d / (D_m / -8.0)))) / (d * l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2.3e+49:
		tmp = w0
	else:
		tmp = (M_m * ((M_m * (w0 * (D_m * h))) / (d / (D_m / -8.0)))) / (d * l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.3e+49)
		tmp = w0;
	else
		tmp = Float64(Float64(M_m * Float64(Float64(M_m * Float64(w0 * Float64(D_m * h))) / Float64(d / Float64(D_m / -8.0)))) / Float64(d * l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.3e+49)
		tmp = w0;
	else
		tmp = (M_m * ((M_m * (w0 * (D_m * h))) / (d / (D_m / -8.0)))) / (d * l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.3e+49], w0, N[(N[(M$95$m * N[(N[(M$95$m * N[(w0 * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D$95$m / -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.30000000000000002e49

   1. Initial program 80.1%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.7%

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

   if 2.30000000000000002e49 < M

   1. Initial program 73.2%

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

   3. Simplified 66.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in l around 0

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 24.6%

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

   10. Simplified 24.6%

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

       1. associate-*l* N/A

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

       2. associate-/r* N/A

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

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

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

   12. Applied egg-rr 32.2%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

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

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

       14. /-lowering-/.f64 25.6%

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

   14. Applied egg-rr 25.6%

       \[\leadsto \frac{\color{blue}{\frac{M \cdot \left(w0 \cdot \left(h \cdot D\right)\right)}{\frac{d}{\frac{D}{-8}}} \cdot M}}{d \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{M \cdot \frac{M \cdot \left(w0 \cdot \left(D \cdot h\right)\right)}{\frac{d}{\frac{D}{-8}}}}{d \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(w0 \cdot D\_m\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d} \cdot \frac{D\_m}{d \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2.2e+52)
   w0
   (* (/ (* (* w0 D_m) (* -0.125 (* h (* M_m M_m)))) d) (/ D_m (* d l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.2e+52) {
		tmp = w0;
	} else {
		tmp = (((w0 * D_m) * (-0.125 * (h * (M_m * M_m)))) / d) * (D_m / (d * l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2.2d+52) then
        tmp = w0
    else
        tmp = (((w0 * d_m) * ((-0.125d0) * (h * (m_m * m_m)))) / d) * (d_m / (d * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.2e+52) {
		tmp = w0;
	} else {
		tmp = (((w0 * D_m) * (-0.125 * (h * (M_m * M_m)))) / d) * (D_m / (d * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2.2e+52:
		tmp = w0
	else:
		tmp = (((w0 * D_m) * (-0.125 * (h * (M_m * M_m)))) / d) * (D_m / (d * l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.2e+52)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(Float64(w0 * D_m) * Float64(-0.125 * Float64(h * Float64(M_m * M_m)))) / d) * Float64(D_m / Float64(d * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.2e+52)
		tmp = w0;
	else
		tmp = (((w0 * D_m) * (-0.125 * (h * (M_m * M_m)))) / d) * (D_m / (d * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.2e+52], w0, N[(N[(N[(N[(w0 * D$95$m), $MachinePrecision] * N[(-0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2e52

   1. Initial program 80.1%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.7%

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

   if 2.2e52 < M

   1. Initial program 73.2%

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

   3. Simplified 66.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in l around 0

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 24.6%

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

   10. Simplified 24.6%

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

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

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

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

   12. Applied egg-rr 32.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(w0 \cdot D\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d} \cdot \frac{D}{d \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.25 \cdot 10^{+49}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{d} \cdot \frac{\left(h \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(w0 \cdot \left(D\_m \cdot D\_m\right)\right)}{d \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2.25e+49)
   w0
   (* (/ -0.125 d) (/ (* (* h (* M_m M_m)) (* w0 (* D_m D_m))) (* d l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e+49) {
		tmp = w0;
	} else {
		tmp = (-0.125 / d) * (((h * (M_m * M_m)) * (w0 * (D_m * D_m))) / (d * l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2.25d+49) then
        tmp = w0
    else
        tmp = ((-0.125d0) / d) * (((h * (m_m * m_m)) * (w0 * (d_m * d_m))) / (d * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e+49) {
		tmp = w0;
	} else {
		tmp = (-0.125 / d) * (((h * (M_m * M_m)) * (w0 * (D_m * D_m))) / (d * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2.25e+49:
		tmp = w0
	else:
		tmp = (-0.125 / d) * (((h * (M_m * M_m)) * (w0 * (D_m * D_m))) / (d * l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.25e+49)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 / d) * Float64(Float64(Float64(h * Float64(M_m * M_m)) * Float64(w0 * Float64(D_m * D_m))) / Float64(d * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.25e+49)
		tmp = w0;
	else
		tmp = (-0.125 / d) * (((h * (M_m * M_m)) * (w0 * (D_m * D_m))) / (d * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.25e+49], w0, N[(N[(-0.125 / d), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.24999999999999991e49

   1. Initial program 80.1%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.7%

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

   if 2.24999999999999991e49 < M

   1. Initial program 73.2%

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

   3. Simplified 66.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in l around 0

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 24.6%

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

   10. Simplified 24.6%

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

       1. associate-*l* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

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

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

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

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

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

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

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

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

       16. *-lowering-*.f64 22.4%

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

   12. Applied egg-rr 22.4%

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

Alternative 9: 63.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{d}{\frac{d}{w0}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 4.4e-209) (/ d (/ d w0)) w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4.4e-209) {
		tmp = d / (d / w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 4.4d-209) then
        tmp = d / (d / w0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4.4e-209) {
		tmp = d / (d / w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 4.4e-209:
		tmp = d / (d / w0)
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 4.4e-209)
		tmp = Float64(d / Float64(d / w0));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 4.4e-209)
		tmp = d / (d / w0);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 4.4e-209], N[(d / N[(d / w0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 4.40000000000000019e-209

   1. Initial program 76.4%

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

   3. Simplified 80.1%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in d around 0

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

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

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

   10. Simplified 29.8%

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

   11. Taylor expanded in d around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({d}^{2} \cdot w0\right)}, \mathsf{*.f64}\left(d, d\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(w0 \cdot {d}^{2}\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left({d}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left(d \cdot d\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       4. *-lowering-*.f64 20.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   13. Simplified 20.2%

       \[\leadsto \frac{\color{blue}{w0 \cdot \left(d \cdot d\right)}}{d \cdot d} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(d \cdot d\right) \cdot w0}{\color{blue}{d} \cdot d} \]

       2. times-frac N/A

          \[\leadsto \frac{d \cdot d}{d} \cdot \color{blue}{\frac{w0}{d}} \]

       3. clear-num N/A

          \[\leadsto \frac{d \cdot d}{d} \cdot \frac{1}{\color{blue}{\frac{d}{w0}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{d \cdot d}{d}}{\color{blue}{\frac{d}{w0}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{d \cdot \frac{d}{d}}{\frac{\color{blue}{d}}{w0}} \]

       6. *-inverses N/A

          \[\leadsto \frac{d \cdot 1}{\frac{d}{w0}} \]

       7. *-rgt-identity N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(d, \color{blue}{\left(\frac{d}{w0}\right)}\right) \]

       9. /-lowering-/.f64 57.4%

          \[\leadsto \mathsf{/.f64}\left(d, \mathsf{/.f64}\left(d, \color{blue}{w0}\right)\right) \]

   15. Applied egg-rr 57.4%

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

   if 4.40000000000000019e-209 < d

   1. Initial program 81.5%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 76.7%

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

Alternative 10: 63.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.6 \cdot 10^{-199}:\\ \;\;\;\;d \cdot \frac{w0}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 4.6e-199) (* d (/ w0 d)) w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4.6e-199) {
		tmp = d * (w0 / d);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 4.6d-199) then
        tmp = d * (w0 / d)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 4.6e-199) {
		tmp = d * (w0 / d);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 4.6e-199:
		tmp = d * (w0 / d)
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 4.6e-199)
		tmp = Float64(d * Float64(w0 / d));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 4.6e-199)
		tmp = d * (w0 / d);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 4.6e-199], N[(d * N[(w0 / d), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 4.6000000000000003e-199

   1. Initial program 76.3%

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

   3. Simplified 80.0%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right), \left({\color{blue}{d}}^{2}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right), \left({d}^{2}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), \left({d}^{2}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left({d}^{2}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \left(d \cdot \color{blue}{d}\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 41.5%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right)\right) \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{w0 + \frac{-0.125}{\ell} \cdot \left(\left(D \cdot D\right) \cdot \frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}\right)} \]

   8. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\ell} + {d}^{2} \cdot w0}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\ell} + {d}^{2} \cdot w0\right), \color{blue}{\left({d}^{2}\right)}\right) \]

   10. Simplified 29.3%

       \[\leadsto \color{blue}{\frac{\left(d \cdot d\right) \cdot w0 + -0.125 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot D\right) \cdot D}{\ell}}{d \cdot d}} \]

   11. Taylor expanded in d around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({d}^{2} \cdot w0\right)}, \mathsf{*.f64}\left(d, d\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(w0 \cdot {d}^{2}\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left({d}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{d}, d\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left(d \cdot d\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

       4. *-lowering-*.f64 19.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{*.f64}\left(d, d\right)\right) \]

   13. Simplified 19.8%

       \[\leadsto \frac{\color{blue}{w0 \cdot \left(d \cdot d\right)}}{d \cdot d} \]
   14. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{w0}{d} \cdot \color{blue}{\frac{d \cdot d}{d}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{w0}{d} \cdot \left(d \cdot \color{blue}{\frac{d}{d}}\right) \]

       3. *-inverses N/A

          \[\leadsto \frac{w0}{d} \cdot \left(d \cdot 1\right) \]

       4. *-rgt-identity N/A

          \[\leadsto \frac{w0}{d} \cdot d \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{w0}{d}\right), \color{blue}{d}\right) \]

       6. /-lowering-/.f64 58.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(w0, d\right), d\right) \]

   15. Applied egg-rr 58.7%

       \[\leadsto \color{blue}{\frac{w0}{d} \cdot d} \]

   if 4.6000000000000003e-199 < d

   1. Initial program 82.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 76.9%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.6 \cdot 10^{-199}:\\ \;\;\;\;d \cdot \frac{w0}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 78.7%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 79.3%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 68.3%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))))))