Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.4%

Time: 19.8s

Alternatives: 7

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: 81.4% 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: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D\_m \cdot \frac{M\_m}{2 \cdot d\_m}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m} \cdot \left(0.5 \cdot \left(D\_m \cdot \frac{h \cdot \frac{M\_m}{d\_m}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1.5e+28)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0) h) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ (* D_m (* M_m 0.5)) d_m)
       (* 0.5 (* D_m (/ (* h (/ M_m d_m)) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28) {
		tmp = w0 * sqrt((1.0 - ((pow((D_m * (M_m / (2.0 * d_m))), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 1.5d+28) then
        tmp = w0 * sqrt((1.0d0 - ((((d_m * (m_m / (2.0d0 * d_m_1))) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m * (m_m * 0.5d0)) / d_m_1) * (0.5d0 * (d_m * ((h * (m_m / d_m_1)) / l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1.5e+28)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) * Float64(0.5 * Float64(D_m * Float64(Float64(h * Float64(M_m / d_m)) / l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28)
		tmp = w0 * sqrt((1.0 - ((((D_m * (M_m / (2.0 * d_m))) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1.5e+28], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.5e28

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 92.8%

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

      2. add-sqr-sqrt 92.8%

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

      3. pow2 92.8%

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

      4. unpow2 92.8%

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

      5. sqrt-prod 55.2%

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

      6. add-sqr-sqrt 92.8%

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

      7. div-inv 92.8%

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

      8. frac-times 92.8%

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

      9. *-commutative 92.8%

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

      10. *-un-lft-identity 92.8%

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

   6. Applied egg-rr 92.8%

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

   if 1.5e28 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 63.8%

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

   3. Simplified 61.6%

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

      1. associate-*r/ 59.5%

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

      2. add-sqr-sqrt 59.5%

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

      3. pow2 59.5%

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

      4. unpow2 59.5%

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

      5. sqrt-prod 59.4%

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

      6. add-sqr-sqrt 59.5%

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

      7. div-inv 59.5%

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

      8. frac-times 59.5%

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

      9. *-commutative 59.5%

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

      10. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. associate-*r/ 61.6%

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

      2. unpow2 61.6%

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

      3. associate-*l* 64.1%

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

      4. *-commutative 64.1%

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

      5. associate-/r* 64.1%

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

      6. associate-*l/ 64.1%

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

      7. div-inv 64.1%

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

      8. metadata-eval 64.1%

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

      9. *-commutative 64.1%

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

      10. associate-/r* 64.1%

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

      11. associate-*l/ 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

   8. Applied egg-rr 66.3%

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

   9. Taylor expanded in M around 0 64.2%

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

       1. associate-/l* 64.2%

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

       2. *-commutative 64.2%

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

       3. *-commutative 64.2%

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

       4. times-frac 61.8%

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

   11. Simplified 61.8%

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

       1. associate-*l/ 64.2%

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

   13. Applied egg-rr 64.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(M \cdot 0.5\right)}{d} \cdot \left(0.5 \cdot \left(D \cdot \frac{h \cdot \frac{M}{d}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m}\\ \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{t\_0}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(0.5 \cdot \left(D\_m \cdot \frac{h \cdot \frac{M\_m}{d\_m}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m (* M_m 0.5)) d_m)))
   (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1.5e+28)
     (* w0 (sqrt (- 1.0 (* h (/ (pow t_0 2.0) l)))))
     (* w0 (sqrt (- 1.0 (* t_0 (* 0.5 (* D_m (/ (* h (/ M_m d_m)) l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m * (M_m * 0.5)) / d_m;
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28) {
		tmp = w0 * sqrt((1.0 - (h * (pow(t_0, 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m * (m_m * 0.5d0)) / d_m_1
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 1.5d+28) then
        tmp = w0 * sqrt((1.0d0 - (h * ((t_0 ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (0.5d0 * (d_m * ((h * (m_m / d_m_1)) / l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m * (M_m * 0.5)) / d_m;
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow(t_0, 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (D_m * (M_m * 0.5)) / d_m
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow(t_0, 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1.5e+28)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((t_0 ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(0.5 * Float64(D_m * Float64(Float64(h * Float64(M_m / d_m)) / l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (D_m * (M_m * 0.5)) / d_m;
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1.5e+28)
		tmp = w0 * sqrt((1.0 - (h * ((t_0 ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (0.5 * (D_m * ((h * (M_m / d_m)) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1.5e+28], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(0.5 * N[(D$95$m * N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.5e28

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 92.8%

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

      2. add-sqr-sqrt 92.8%

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

      3. pow2 92.8%

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

      4. unpow2 92.8%

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

      5. sqrt-prod 55.2%

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

      6. add-sqr-sqrt 92.8%

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

      7. div-inv 92.8%

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

      8. frac-times 92.8%

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

      9. *-commutative 92.8%

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

      10. *-un-lft-identity 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-/l* 91.4%

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

      3. *-commutative 91.4%

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

      4. associate-/r* 91.4%

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

      5. associate-*l/ 91.5%

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

      6. div-inv 91.5%

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

      7. metadata-eval 91.5%

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

   8. Applied egg-rr 91.5%

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

   if 1.5e28 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 63.8%

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

   3. Simplified 61.6%

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

      1. associate-*r/ 59.5%

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

      2. add-sqr-sqrt 59.5%

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

      3. pow2 59.5%

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

      4. unpow2 59.5%

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

      5. sqrt-prod 59.4%

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

      6. add-sqr-sqrt 59.5%

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

      7. div-inv 59.5%

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

      8. frac-times 59.5%

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

      9. *-commutative 59.5%

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

      10. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. associate-*r/ 61.6%

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

      2. unpow2 61.6%

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

      3. associate-*l* 64.1%

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

      4. *-commutative 64.1%

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

      5. associate-/r* 64.1%

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

      6. associate-*l/ 64.1%

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

      7. div-inv 64.1%

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

      8. metadata-eval 64.1%

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

      9. *-commutative 64.1%

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

      10. associate-/r* 64.1%

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

      11. associate-*l/ 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

   8. Applied egg-rr 66.3%

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

   9. Taylor expanded in M around 0 64.2%

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

       1. associate-/l* 64.2%

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

       2. *-commutative 64.2%

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

       3. *-commutative 64.2%

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

       4. times-frac 61.8%

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

   11. Simplified 61.8%

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

       1. associate-*l/ 64.2%

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

   13. Applied egg-rr 64.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 1.5 \cdot 10^{+28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(M \cdot 0.5\right)}{d} \cdot \left(0.5 \cdot \left(D \cdot \frac{h \cdot \frac{M}{d}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-193}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \frac{M\_m}{2 \cdot d\_m}\right) \cdot \left(\frac{M\_m}{d\_m} \cdot \left(\frac{h}{\ell} \cdot \left(D\_m \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 2e-193)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* D_m (/ M_m (* 2.0 d_m)))
       (* (/ M_m d_m) (* (/ h l) (* D_m 0.5)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2e-193) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m * (M_m / (2.0 * d_m))) * ((M_m / d_m) * ((h / l) * (D_m * 0.5))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 2d-193) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((d_m * (m_m / (2.0d0 * d_m_1))) * ((m_m / d_m_1) * ((h / l) * (d_m * 0.5d0))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2e-193) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * (M_m / (2.0 * d_m))) * ((M_m / d_m) * ((h / l) * (D_m * 0.5))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 2e-193:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((D_m * (M_m / (2.0 * d_m))) * ((M_m / d_m) * ((h / l) * (D_m * 0.5))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2e-193)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) * Float64(Float64(M_m / d_m) * Float64(Float64(h / l) * Float64(D_m * 0.5)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 2e-193)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((D_m * (M_m / (2.0 * d_m))) * ((M_m / d_m) * ((h / l) * (D_m * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 2e-193], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.0000000000000001e-193

   1. Initial program 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in D around 0 75.4%

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

   if 2.0000000000000001e-193 < M

   1. Initial program 73.0%

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

   3. Simplified 72.0%

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

      1. associate-*r/ 80.3%

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

      2. add-sqr-sqrt 80.3%

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

      3. pow2 80.3%

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

      4. unpow2 80.3%

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

      5. sqrt-prod 47.8%

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

      6. add-sqr-sqrt 80.3%

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

      7. div-inv 80.3%

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

      8. frac-times 80.3%

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

      9. *-commutative 80.3%

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

      10. *-un-lft-identity 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. associate-*r/ 72.0%

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

      2. unpow2 72.0%

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

      3. associate-*l* 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-/r* 73.1%

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

      6. associate-*l/ 73.1%

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

      7. div-inv 73.1%

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

      8. metadata-eval 73.1%

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

      9. *-commutative 73.1%

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

      10. associate-/r* 73.1%

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

      11. associate-*l/ 74.1%

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

      12. div-inv 74.1%

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

      13. metadata-eval 74.1%

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

   8. Applied egg-rr 74.1%

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

   9. Taylor expanded in M around 0 76.2%

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

       1. associate-/l* 72.2%

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

       2. *-commutative 72.2%

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

       3. *-commutative 72.2%

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

       4. times-frac 72.1%

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

   11. Simplified 72.1%

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

       1. *-un-lft-identity 72.1%

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

       2. metadata-eval 72.1%

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

       3. div-inv 72.1%

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

       4. *-commutative 72.1%

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

       5. associate-*l/ 72.1%

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

       6. frac-times 72.1%

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

       7. associate-*r* 72.1%

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

   13. Applied egg-rr 72.1%

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

       1. *-lft-identity 72.1%

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

       2. associate-/l* 72.1%

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

       3. *-commutative 72.1%

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

       4. associate-*r* 73.1%

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

       5. *-commutative 73.1%

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

   15. Simplified 73.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-193}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 4 \cdot 10^{-21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m} \cdot \left(0.5 \cdot \left(D\_m \cdot \left(\frac{M\_m}{d\_m} \cdot \frac{h}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 4e-21)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ (* D_m (* M_m 0.5)) d_m)
       (* 0.5 (* D_m (* (/ M_m d_m) (/ h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 4e-21) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((M_m / d_m) * (h / l)))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 4d-21) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m * (m_m * 0.5d0)) / d_m_1) * (0.5d0 * (d_m * ((m_m / d_m_1) * (h / l)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 4e-21) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((M_m / d_m) * (h / l)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 4e-21:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((M_m / d_m) * (h / l)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 4e-21)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) * Float64(0.5 * Float64(D_m * Float64(Float64(M_m / d_m) * Float64(h / l))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 4e-21)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (((D_m * (M_m * 0.5)) / d_m) * (0.5 * (D_m * ((M_m / d_m) * (h / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 4e-21], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.99999999999999963e-21

   1. Initial program 80.8%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in D around 0 75.5%

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

   if 3.99999999999999963e-21 < D

   1. Initial program 77.0%

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

   3. Simplified 77.0%

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

      1. associate-*r/ 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. pow2 78.7%

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

      4. unpow2 78.7%

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

      5. sqrt-prod 43.8%

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

      6. add-sqr-sqrt 78.7%

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

      7. div-inv 78.7%

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

      8. frac-times 78.7%

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

      9. *-commutative 78.7%

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

      10. *-un-lft-identity 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. associate-*r/ 77.0%

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

      2. unpow2 77.0%

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

      3. associate-*l* 77.2%

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

      4. *-commutative 77.2%

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

      5. associate-/r* 77.2%

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

      6. associate-*l/ 77.1%

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

      7. div-inv 77.1%

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

      8. metadata-eval 77.1%

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

      9. *-commutative 77.1%

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

      10. associate-/r* 77.1%

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

      11. associate-*l/ 77.1%

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

      12. div-inv 77.1%

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

      13. metadata-eval 77.1%

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

   8. Applied egg-rr 77.1%

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

   9. Taylor expanded in M around 0 76.8%

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

       1. associate-/l* 81.1%

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

       2. *-commutative 81.1%

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

       3. *-commutative 81.1%

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

       4. times-frac 77.1%

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

   11. Simplified 77.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{-21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(M \cdot 0.5\right)}{d} \cdot \left(0.5 \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot {d\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 1.9e-52)
   w0
   (+
    w0
    (/
     (* -0.125 (* (* (* M_m D_m) (* M_m D_m)) (* w0 h)))
     (* l (pow d_m 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.9e-52) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * pow(d_m, 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 1.9d-52) then
        tmp = w0
    else
        tmp = w0 + (((-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * (w0 * h))) / (l * (d_m_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 1.9e-52) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * Math.pow(d_m, 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 1.9e-52:
		tmp = w0
	else:
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * math.pow(d_m, 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 1.9e-52)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(w0 * h))) / Float64(l * (d_m ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 1.9e-52)
		tmp = w0;
	else
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * (d_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 1.9e-52], w0, N[(w0 + N[(N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.9000000000000002e-52

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in D around 0 74.8%

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

   if 1.9000000000000002e-52 < M

   1. Initial program 72.4%

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

   3. Simplified 70.8%

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

      1. add-sqr-sqrt 34.4%

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

      2. sqrt-unprod 25.2%

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

      3. *-commutative 25.2%

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

      4. *-commutative 25.2%

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

      5. swap-sqr 20.3%

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

   6. Applied egg-rr 20.3%

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

   7. Taylor expanded in D around 0 39.7%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ 39.7%

         \[\leadsto w0 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. associate-*r* 41.6%

         \[\leadsto w0 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)\right)}}{{d}^{2} \cdot \ell} \]

      3. unpow2 41.6%

         \[\leadsto w0 + \frac{-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \]

      4. unpow2 41.6%

         \[\leadsto w0 + \frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \]

      5. swap-sqr 45.1%

         \[\leadsto w0 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \]

      6. unpow2 45.1%

         \[\leadsto w0 + \frac{-0.125 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \]

   9. Simplified 45.1%

      \[\leadsto \color{blue}{w0 + \frac{-0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   10. Step-by-step derivation

       1. unpow2 45.1%

          \[\leadsto w0 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \]

   11. Applied egg-rr 45.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

3. Simplified 79.5%

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

   1. associate-*r/ 87.3%

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

   2. add-sqr-sqrt 87.3%

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

   3. pow2 87.3%

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

   4. unpow2 87.3%

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

   5. sqrt-prod 55.9%

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

   6. add-sqr-sqrt 87.3%

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

   7. div-inv 87.3%

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

   8. frac-times 87.3%

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

   9. *-commutative 87.3%

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

   10. *-un-lft-identity 87.3%

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

6. Applied egg-rr 87.3%

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

   1. associate-*r/ 79.5%

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

   2. unpow2 79.5%

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

   3. associate-*l* 80.8%

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

   4. *-commutative 80.8%

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

   5. associate-/r* 80.8%

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

   6. associate-*l/ 80.4%

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

   7. div-inv 80.4%

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

   8. metadata-eval 80.4%

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

   9. *-commutative 80.4%

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

   10. associate-/r* 80.4%

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

   11. associate-*l/ 81.6%

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

   12. div-inv 81.6%

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

   13. metadata-eval 81.6%

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

8. Applied egg-rr 81.6%

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

9. Taylor expanded in M around 0 81.5%

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

    1. associate-/l* 80.4%

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

    2. *-commutative 80.4%

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

    3. *-commutative 80.4%

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

    4. times-frac 79.6%

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

11. Simplified 79.6%

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

    1. associate-*l/ 86.3%

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

13. Applied egg-rr 86.3%

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

14. Final simplification 86.3%

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

Alternative 7: 69.0% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

3. Simplified 79.5%

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

5. Taylor expanded in D around 0 68.6%

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

6. Final simplification 68.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))))