Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.5%

Time: 15.7s

Alternatives: 7

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* M_m (/ (* D_m 0.5) d_m))))
   (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+94)
     (* w0 (sqrt (- 1.0 (* h (/ (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0) l)))))
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_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 t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+94) {
		tmp = w0 * sqrt((1.0 - (h * (pow(((D_m * (M_m * 0.5)) / d_m), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d_m_1)
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+94) then
        tmp = w0 * sqrt((1.0d0 - (h * ((((d_m * (m_m * 0.5d0)) / d_m_1) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    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 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+94) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_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):
	t_0 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+94:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_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)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+94)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_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)
	t_0 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+94)
		tmp = w0 * sqrt((1.0 - (h * ((((D_m * (M_m * 0.5)) / d_m) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+94], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.0000000000000001e94

   1. Initial program 87.2%

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

   3. Simplified 85.5%

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

      1. expm1-log1p-u 70.0%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 70.0%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 85.5%

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

      1. associate--l+ 85.5%

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

      2. metadata-eval 85.5%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-*l/ 91.3%

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

      6. associate-/l* 90.1%

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

      7. associate-*r/ 90.1%

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

      8. associate-/l* 91.8%

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

      9. *-commutative 91.8%

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

      10. associate-*l* 91.8%

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

      11. *-commutative 91.8%

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

      12. associate-/l* 90.5%

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

   8. Simplified 90.5%

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

      1. associate-*r/ 91.8%

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

   10. Applied egg-rr 91.8%

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

   if 5.0000000000000001e94 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 44.3%

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

   3. Simplified 48.0%

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

      1. expm1-log1p-u 7.6%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 7.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 48.1%

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

      1. associate--l+ 48.0%

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

      2. metadata-eval 48.0%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 48.0%

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

      4. *-commutative 48.0%

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

      5. unpow2 48.0%

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

      6. associate-*r* 64.6%

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

      7. associate-*r/ 64.6%

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

      8. associate-*r/ 64.6%

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

   8. Applied egg-rr 64.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+33}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D\_m \cdot \frac{M\_m \cdot 0.5}{d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* M_m (/ (* D_m 0.5) d_m))))
   (if (<= (* M_m D_m) 1e+33)
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* D_m (/ (* M_m 0.5) d_m)) 2.0) l)))))
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_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 t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((M_m * D_m) <= 1e+33) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D_m * ((M_m * 0.5) / d_m)), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d_m_1)
    if ((m_m * d_m) <= 1d+33) then
        tmp = w0 * sqrt((1.0d0 - (h * (((d_m * ((m_m * 0.5d0) / d_m_1)) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    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 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((M_m * D_m) <= 1e+33) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D_m * ((M_m * 0.5) / d_m)), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_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):
	t_0 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if (M_m * D_m) <= 1e+33:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((D_m * ((M_m * 0.5) / d_m)), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_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)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e+33)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(Float64(M_m * 0.5) / d_m)) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_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)
	t_0 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if ((M_m * D_m) <= 1e+33)
		tmp = w0 * sqrt((1.0 - (h * (((D_m * ((M_m * 0.5) / d_m)) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+33], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999995e32

   1. Initial program 84.0%

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

   3. Simplified 83.1%

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

      1. expm1-log1p-u 67.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 67.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 83.1%

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

      1. associate--l+ 83.1%

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

      2. metadata-eval 83.1%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 83.1%

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

      4. *-commutative 83.1%

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

      5. associate-*l/ 89.4%

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

      6. associate-/l* 88.0%

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

      7. associate-*r/ 88.0%

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

      8. associate-/l* 89.0%

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

      9. *-commutative 89.0%

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

      10. associate-*l* 89.0%

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

      11. *-commutative 89.0%

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

      12. associate-/l* 88.0%

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

   8. Simplified 88.0%

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

   if 9.9999999999999995e32 < (*.f64 M D)

   1. Initial program 76.8%

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

   3. Simplified 74.7%

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

      1. expm1-log1p-u 43.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 43.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 74.7%

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

      1. associate--l+ 74.7%

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

      2. metadata-eval 74.7%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 74.7%

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

      4. *-commutative 74.7%

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

      5. unpow2 74.7%

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

      6. associate-*r* 82.5%

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

      7. associate-*r/ 82.5%

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

      8. associate-*r/ 82.5%

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

   8. Applied egg-rr 82.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{+33}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ (* D_m 0.5) d_m))))
   (if (<= (/ h l) -1e-321) (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_0))))) w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((h / l) <= -1e-321) {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d_m_1)
    if ((h / l) <= (-1d-321)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((h / l) <= -1e-321) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if (h / l) <= -1e-321:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (Float64(h / l) <= -1e-321)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if ((h / l) <= -1e-321)
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e-321], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.98013e-322

   1. Initial program 77.1%

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

   3. Simplified 77.2%

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

      1. expm1-log1p-u 46.0%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 46.0%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 77.2%

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

      1. associate--l+ 77.2%

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

      2. metadata-eval 77.2%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 77.2%

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

      4. *-commutative 77.2%

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

      5. unpow2 77.2%

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

      6. associate-*r* 78.1%

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

      7. associate-*r/ 78.1%

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

      8. associate-*r/ 78.1%

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

   8. Applied egg-rr 78.1%

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

   if -9.98013e-322 < (/.f64 h l)

   1. Initial program 90.7%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in M around 0 99.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in w0 around 0 99.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D\_m \cdot \frac{0.5}{d\_m}\right)\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (/ 0.5 d_m)))))
   (if (<= (/ h l) -1e-255) (* w0 (sqrt (- 1.0 (* (/ h l) (* t_0 t_0))))) w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m * (0.5 / d_m));
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_m * (0.5d0 / d_m_1))
    if ((h / l) <= (-1d-255)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (t_0 * t_0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m * (0.5 / d_m));
	double tmp;
	if ((h / l) <= -1e-255) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = M_m * (D_m * (0.5 / d_m))
	tmp = 0
	if (h / l) <= -1e-255:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (t_0 * t_0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(0.5 / d_m)))
	tmp = 0.0
	if (Float64(h / l) <= -1e-255)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = M_m * (D_m * (0.5 / d_m));
	tmp = 0.0;
	if ((h / l) <= -1e-255)
		tmp = w0 * sqrt((1.0 - ((h / l) * (t_0 * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e-255], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1e-255

   1. Initial program 77.5%

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

   3. Simplified 76.9%

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

      1. unpow-prod-down 60.0%

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

      2. add-sqr-sqrt 59.9%

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

      3. unpow-prod-down 60.0%

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

      4. sqrt-pow1 48.0%

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

      5. metadata-eval 48.0%

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

      6. unpow-prod-down 48.0%

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

      7. pow1 48.0%

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

      8. div-inv 48.0%

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

      9. pow1 48.0%

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

      10. associate-/r* 48.0%

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

      11. metadata-eval 48.0%

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

      12. unpow-prod-down 58.8%

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

      13. sqrt-pow1 76.9%

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

      14. metadata-eval 76.9%

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

      15. unpow-prod-down 76.9%

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

      16. pow1 76.9%

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

      17. div-inv 76.9%

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

      18. pow1 76.9%

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

      19. associate-/r* 76.9%

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

      20. metadata-eval 76.9%

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

   6. Applied egg-rr 76.9%

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

   if -1e-255 < (/.f64 h l)

   1. Initial program 89.1%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in M around 0 97.0%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in w0 around 0 97.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ \mathbf{if}\;M\_m \leq 6.5 \cdot 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \left(D\_m \cdot 0.5\right)\right)}{\ell \cdot \left(2 \cdot \left(d\_m \cdot \frac{\frac{d\_m}{M\_m}}{D\_m}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* M_m (/ (* D_m 0.5) d_m))))
   (if (<= M_m 6.5e-120)
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (* h (* M_m (* D_m 0.5)))
         (* l (* 2.0 (* d_m (/ (/ d_m M_m) D_m))))))))
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_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 t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (M_m <= 6.5e-120) {
		tmp = w0 * sqrt((1.0 - ((h * (M_m * (D_m * 0.5))) / (l * (2.0 * (d_m * ((d_m / M_m) / D_m)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d_m_1)
    if (m_m <= 6.5d-120) then
        tmp = w0 * sqrt((1.0d0 - ((h * (m_m * (d_m * 0.5d0))) / (l * (2.0d0 * (d_m_1 * ((d_m_1 / m_m) / d_m)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    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 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (M_m <= 6.5e-120) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (M_m * (D_m * 0.5))) / (l * (2.0 * (d_m * ((d_m / M_m) / D_m)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_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):
	t_0 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if M_m <= 6.5e-120:
		tmp = w0 * math.sqrt((1.0 - ((h * (M_m * (D_m * 0.5))) / (l * (2.0 * (d_m * ((d_m / M_m) / D_m)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_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)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (M_m <= 6.5e-120)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(D_m * 0.5))) / Float64(l * Float64(2.0 * Float64(d_m * Float64(Float64(d_m / M_m) / D_m))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_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)
	t_0 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if (M_m <= 6.5e-120)
		tmp = w0 * sqrt((1.0 - ((h * (M_m * (D_m * 0.5))) / (l * (2.0 * (d_m * ((d_m / M_m) / D_m)))))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_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_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.5e-120], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(2.0 * N[(d$95$m * N[(N[(d$95$m / M$95$m), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 6.50000000000000029e-120

   1. Initial program 85.4%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 85.4%

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

      2. unpow2 85.4%

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

      3. associate-/r* 85.4%

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

      4. clear-num 85.4%

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

      5. frac-times 85.4%

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

      6. *-un-lft-identity 85.4%

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

      7. associate-/l* 85.4%

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

      8. div-inv 85.4%

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

      9. metadata-eval 85.4%

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

      10. associate-/l* 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. frac-times 89.0%

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

      2. associate-*l* 89.0%

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

      3. associate-/r* 88.4%

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

   8. Applied egg-rr 88.4%

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

   if 6.50000000000000029e-120 < M

   1. Initial program 78.1%

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

   3. Simplified 80.3%

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

      1. expm1-log1p-u 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      2. expm1-undefine 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   6. Applied egg-rr 80.3%

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

      1. associate--l+ 80.3%

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

      2. metadata-eval 80.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 80.3%

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

      4. *-commutative 80.3%

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

      5. unpow2 80.3%

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

      6. associate-*r* 84.0%

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

      7. associate-*r/ 84.0%

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

      8. associate-*r/ 84.0%

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

   8. Applied egg-rr 84.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(D \cdot 0.5\right)\right)}{\ell \cdot \left(2 \cdot \left(d \cdot \frac{\frac{d}{M}}{D}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 6.6 \cdot 10^{+59}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m \cdot \left(2 \cdot \frac{d\_m}{M\_m \cdot D\_m}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 6.6e+59)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ h l)
       (/ (* M_m (* D_m 0.5)) (* d_m (* 2.0 (/ d_m (* M_m D_m)))))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 6.6e+59) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * (D_m * 0.5)) / (d_m * (2.0 * (d_m / (M_m * D_m))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 6.6d+59) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m_m * (d_m * 0.5d0)) / (d_m_1 * (2.0d0 * (d_m_1 / (m_m * d_m))))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 6.6e+59) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((M_m * (D_m * 0.5)) / (d_m * (2.0 * (d_m / (M_m * D_m))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 6.6e+59:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((M_m * (D_m * 0.5)) / (d_m * (2.0 * (d_m / (M_m * D_m))))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 6.6e+59)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M_m * Float64(D_m * 0.5)) / Float64(d_m * Float64(2.0 * Float64(d_m / Float64(M_m * D_m)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 6.6e+59)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * (D_m * 0.5)) / (d_m * (2.0 * (d_m / (M_m * D_m))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 6.6e+59], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(2.0 * N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 6.5999999999999999e59

   1. Initial program 84.0%

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

   3. Simplified 82.0%

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

      1. associate-*r/ 84.0%

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

      2. unpow2 84.0%

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

      3. associate-/r* 84.0%

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

      4. clear-num 84.0%

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

      5. frac-times 84.0%

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

      6. *-un-lft-identity 84.0%

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

      7. associate-/l* 84.0%

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

      8. div-inv 84.0%

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

      9. metadata-eval 84.0%

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

      10. associate-/l* 84.0%

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

   6. Applied egg-rr 84.0%

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

   if 6.5999999999999999e59 < d

   1. Initial program 79.2%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in M around 0 89.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in w0 around 0 89.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.6 \cdot 10^{+59}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot \left(2 \cdot \frac{d}{M \cdot D}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% 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 82.9%

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

3. Simplified 81.7%

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

5. Taylor expanded in M around 0 72.0%

   \[\leadsto w0 \cdot \color{blue}{1} \]

6. Taylor expanded in w0 around 0 72.0%

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

Reproduce

herbie shell --seed 2024111 
(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))))))