Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 88.7%

Time: 15.0s

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: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{D\_m}{d}\\ \mathbf{if}\;M\_m \leq 1.5 \cdot 10^{-149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right) \cdot \left(\left(0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)\right) \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot t\_0\right) \cdot \left(M\_m \cdot \frac{h \cdot t\_0}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ D_m d))))
   (if (<= M_m 1.5e-149)
     (*
      w0
      (sqrt
       (-
        1.0
        (/ (* (* 0.5 (/ (* M_m D_m) d)) (* (* 0.5 (* D_m (/ M_m d))) h)) l))))
     (* w0 (sqrt (- 1.0 (* (* M_m t_0) (* M_m (/ (* h t_0) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m / d);
	double tmp;
	if (M_m <= 1.5e-149) {
		tmp = w0 * sqrt((1.0 - (((0.5 * ((M_m * D_m) / d)) * ((0.5 * (D_m * (M_m / d))) * h)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (d_m / d)
    if (m_m <= 1.5d-149) then
        tmp = w0 * sqrt((1.0d0 - (((0.5d0 * ((m_m * d_m) / d)) * ((0.5d0 * (d_m * (m_m / d))) * h)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * t_0) * (m_m * ((h * t_0) / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m / d);
	double tmp;
	if (M_m <= 1.5e-149) {
		tmp = w0 * Math.sqrt((1.0 - (((0.5 * ((M_m * D_m) / d)) * ((0.5 * (D_m * (M_m / d))) * h)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (D_m / d)
	tmp = 0
	if M_m <= 1.5e-149:
		tmp = w0 * math.sqrt((1.0 - (((0.5 * ((M_m * D_m) / d)) * ((0.5 * (D_m * (M_m / d))) * h)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(D_m / d))
	tmp = 0.0
	if (M_m <= 1.5e-149)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(Float64(M_m * D_m) / d)) * Float64(Float64(0.5 * Float64(D_m * Float64(M_m / d))) * h)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * t_0) * Float64(M_m * Float64(Float64(h * t_0) / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (D_m / d);
	tmp = 0.0;
	if (M_m <= 1.5e-149)
		tmp = w0 * sqrt((1.0 - (((0.5 * ((M_m * D_m) / d)) * ((0.5 * (D_m * (M_m / d))) * h)) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.5e-149], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * t$95$0), $MachinePrecision] * N[(M$95$m * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.5000000000000001e-149

   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%

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

      1. unpow2 84.1%

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

      2. associate-*r/ 83.4%

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

      3. div-inv 83.5%

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

      4. associate-*r* 82.2%

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

      5. *-un-lft-identity 82.2%

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

      6. times-frac 82.2%

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

      7. metadata-eval 82.2%

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

      8. associate-/r* 82.2%

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

      9. metadata-eval 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-*r/ 86.2%

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

      2. associate-*l* 87.4%

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

      3. associate-*r* 87.4%

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

      4. *-commutative 87.4%

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

   8. Applied egg-rr 87.4%

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

      1. pow1 87.4%

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

      2. associate-*l* 89.3%

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

      3. associate-*r* 89.3%

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

      4. associate-*l/ 89.3%

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

      5. *-commutative 89.3%

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

      6. associate-*r/ 89.3%

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

      7. associate-/l* 89.2%

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

   10. Applied egg-rr 89.2%

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

   11. Taylor expanded in M around 0 89.2%

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

   if 1.5000000000000001e-149 < M

   1. Initial program 78.5%

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

   3. Simplified 78.5%

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

      1. associate-*r/ 78.5%

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

      2. associate-*r/ 79.7%

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

      3. associate-*r/ 79.7%

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

      4. unpow2 79.7%

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

      5. unpow2 79.7%

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

      6. add-sqr-sqrt 79.7%

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

      7. pow2 79.7%

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

      8. sqrt-pow1 79.7%

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

      9. metadata-eval 79.7%

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

      10. pow1 79.7%

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

      11. *-un-lft-identity 79.7%

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

      12. times-frac 79.7%

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

      13. metadata-eval 79.7%

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

   6. Applied egg-rr 79.7%

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

      1. add-sqr-sqrt 57.2%

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

      2. pow2 57.2%

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

      3. sqrt-prod 38.1%

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

      4. sqrt-pow1 42.0%

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

      5. metadata-eval 42.0%

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

      6. pow1 42.0%

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

      7. associate-*r* 42.0%

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

   8. Applied egg-rr 42.0%

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

      1. unpow-prod-down 38.0%

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

      2. associate-*r* 38.0%

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

      3. pow2 38.0%

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

      4. pow2 38.0%

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

      5. add-sqr-sqrt 79.7%

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

      6. associate-*r/ 78.5%

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

      7. associate-*r* 78.5%

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

      8. associate-*r* 78.5%

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

      9. associate-*l* 82.7%

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

      10. associate-*r* 82.7%

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

      11. associate-*r* 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. associate-*r/ 86.6%

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

       2. associate-*r/ 86.6%

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

   12. Applied egg-rr 86.6%

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

       1. associate-*l* 86.6%

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

       2. associate-*r/ 86.3%

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

       3. associate-*r/ 86.3%

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

   14. Simplified 86.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \frac{h \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\\ \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-227}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (* 0.5 (/ D_m d)))))
   (if (<= (* M_m D_m) 1e-227)
     w0
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (0.5 * (D_m / d));
	double tmp;
	if ((M_m * D_m) <= 1e-227) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (0.5d0 * (d_m / d))
    if ((m_m * d_m) <= 1d-227) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (0.5 * (D_m / d));
	double tmp;
	if ((M_m * D_m) <= 1e-227) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = M_m * (0.5 * (D_m / d))
	tmp = 0
	if (M_m * D_m) <= 1e-227:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(0.5 * Float64(D_m / d)))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-227)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = M_m * (0.5 * (D_m / d));
	tmp = 0.0;
	if ((M_m * D_m) <= 1e-227)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-227], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.99999999999999945e-228

   1. Initial program 85.7%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in M around 0 77.6%

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

   if 9.99999999999999945e-228 < (*.f64 M D)

   1. Initial program 73.1%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 73.1%

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

      2. associate-*r/ 70.9%

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

      3. associate-*r/ 70.9%

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

      4. unpow2 70.9%

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

      5. unpow2 70.9%

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

      6. add-sqr-sqrt 70.9%

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

      7. pow2 70.9%

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

      8. sqrt-pow1 70.9%

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

      9. metadata-eval 70.9%

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

      10. pow1 70.9%

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

      11. *-un-lft-identity 70.9%

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

      12. times-frac 70.9%

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

      13. metadata-eval 70.9%

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

   6. Applied egg-rr 70.9%

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

      1. add-sqr-sqrt 49.5%

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

      2. pow2 49.5%

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

      3. sqrt-prod 31.5%

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

      4. sqrt-pow1 34.0%

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

      5. metadata-eval 34.0%

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

      6. pow1 34.0%

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

      7. associate-*r* 34.0%

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

   8. Applied egg-rr 34.0%

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

      1. unpow-prod-down 31.5%

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

      2. associate-*r* 31.5%

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

      3. pow2 31.5%

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

      4. pow2 31.5%

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

      5. add-sqr-sqrt 70.9%

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

      6. associate-*r/ 74.5%

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

      7. associate-*r* 74.5%

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

      8. associate-*r* 74.5%

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

      9. associate-*l* 79.8%

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

      10. associate-*r* 79.8%

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

      11. associate-*r* 79.8%

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

   10. Applied egg-rr 79.8%

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

Alternative 3: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-268}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right) \cdot \left(0.5 \cdot \left(D\_m \cdot \left(\frac{h}{d} \cdot \frac{M\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 5e-268)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (* 0.5 (/ D_m d))) (* 0.5 (* D_m (* (/ h d) (/ M_m l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e-268) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d))) * (0.5 * (D_m * ((h / d) * (M_m / l)))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m_m * d_m) <= 5d-268) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * (0.5d0 * (d_m / d))) * (0.5d0 * (d_m * ((h / d) * (m_m / l)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e-268) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d))) * (0.5 * (D_m * ((h / d) * (M_m / l)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 5e-268:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d))) * (0.5 * (D_m * ((h / d) * (M_m / l)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-268)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d))) * Float64(0.5 * Float64(D_m * Float64(Float64(h / d) * Float64(M_m / l))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 5e-268)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d))) * (0.5 * (D_m * ((h / d) * (M_m / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-268], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(N[(h / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4.9999999999999999e-268

   1. Initial program 86.4%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in M around 0 77.3%

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

   if 4.9999999999999999e-268 < (*.f64 M D)

   1. Initial program 72.7%

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

   3. Simplified 73.9%

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

      1. associate-*r/ 72.7%

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

      2. associate-*r/ 71.9%

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

      3. associate-*r/ 71.9%

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

      4. unpow2 71.9%

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

      5. unpow2 71.9%

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

      6. add-sqr-sqrt 71.9%

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

      7. pow2 71.9%

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

      8. sqrt-pow1 71.9%

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

      9. metadata-eval 71.9%

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

      10. pow1 71.9%

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

      11. *-un-lft-identity 71.9%

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

      12. times-frac 71.9%

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

      13. metadata-eval 71.9%

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

   6. Applied egg-rr 71.9%

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

      1. add-sqr-sqrt 50.8%

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

      2. pow2 50.8%

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

      3. sqrt-prod 31.6%

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

      4. sqrt-pow1 34.0%

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

      5. metadata-eval 34.0%

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

      6. pow1 34.0%

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

      7. associate-*r* 34.0%

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

   8. Applied egg-rr 34.0%

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

      1. unpow-prod-down 31.6%

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

      2. associate-*r* 31.6%

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

      3. pow2 31.6%

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

      4. pow2 31.6%

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

      5. add-sqr-sqrt 71.9%

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

      6. associate-*r/ 73.9%

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

      7. associate-*r* 73.9%

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

      8. associate-*r* 73.9%

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

      9. associate-*l* 78.8%

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

      10. associate-*r* 78.8%

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

      11. associate-*r* 78.8%

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

   10. Applied egg-rr 78.8%

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

   11. Taylor expanded in M around 0 69.8%

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

       1. associate-/l* 69.4%

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

       2. *-commutative 69.4%

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

       3. times-frac 71.4%

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

   13. Simplified 71.4%

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

Alternative 4: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{D\_m}{d}\\ \mathbf{if}\;M\_m \leq 1.75 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot t\_0\right) \cdot \left(M\_m \cdot \frac{h \cdot t\_0}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ D_m d))))
   (if (<= M_m 1.75e-185)
     w0
     (* w0 (sqrt (- 1.0 (* (* M_m t_0) (* M_m (/ (* h t_0) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m / d);
	double tmp;
	if (M_m <= 1.75e-185) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (d_m / d)
    if (m_m <= 1.75d-185) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * t_0) * (m_m * ((h * t_0) / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 0.5 * (D_m / d);
	double tmp;
	if (M_m <= 1.75e-185) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = 0.5 * (D_m / d)
	tmp = 0
	if M_m <= 1.75e-185:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(0.5 * Float64(D_m / d))
	tmp = 0.0
	if (M_m <= 1.75e-185)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * t_0) * Float64(M_m * Float64(Float64(h * t_0) / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = 0.5 * (D_m / d);
	tmp = 0.0;
	if (M_m <= 1.75e-185)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m * t_0) * (M_m * ((h * t_0) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.75e-185], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * t$95$0), $MachinePrecision] * N[(M$95$m * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.7499999999999999e-185

   1. Initial program 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in M around 0 75.2%

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

   if 1.7499999999999999e-185 < M

   1. Initial program 77.6%

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

   3. Simplified 77.7%

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

      1. associate-*r/ 77.6%

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

      2. associate-*r/ 79.7%

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

      3. associate-*r/ 79.8%

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

      4. unpow2 79.8%

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

      5. unpow2 79.8%

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

      6. add-sqr-sqrt 79.8%

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

      7. pow2 79.8%

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

      8. sqrt-pow1 79.8%

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

      9. metadata-eval 79.8%

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

      10. pow1 79.8%

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

      11. *-un-lft-identity 79.8%

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

      12. times-frac 79.8%

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

      13. metadata-eval 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. add-sqr-sqrt 58.4%

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

      2. pow2 58.4%

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

      3. sqrt-prod 39.1%

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

      4. sqrt-pow1 42.9%

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

      5. metadata-eval 42.9%

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

      6. pow1 42.9%

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

      7. associate-*r* 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. unpow-prod-down 39.1%

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

      2. associate-*r* 39.1%

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

      3. pow2 39.1%

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

      4. pow2 39.1%

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

      5. add-sqr-sqrt 79.8%

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

      6. associate-*r/ 77.7%

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

      7. associate-*r* 77.7%

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

      8. associate-*r* 77.7%

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

      9. associate-*l* 81.7%

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

      10. associate-*r* 81.7%

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

      11. associate-*r* 81.7%

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

   10. Applied egg-rr 81.7%

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

       1. associate-*r/ 86.3%

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

       2. associate-*r/ 86.3%

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

   12. Applied egg-rr 86.3%

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

       1. associate-*l* 86.3%

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

       2. associate-*r/ 86.0%

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

       3. associate-*r/ 86.0%

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

   14. Simplified 86.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \frac{h \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

   1. associate-*r/ 81.9%

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

   2. associate-*r/ 84.4%

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

   3. associate-*r/ 84.8%

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

   4. unpow2 84.8%

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

   5. unpow2 84.8%

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

   6. add-sqr-sqrt 84.8%

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

   7. pow2 84.8%

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

   8. sqrt-pow1 84.8%

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

   9. metadata-eval 84.8%

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

   10. pow1 84.8%

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

   11. *-un-lft-identity 84.8%

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

   12. times-frac 84.8%

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

   13. metadata-eval 84.8%

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

6. Applied egg-rr 84.8%

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

   1. add-sqr-sqrt 64.9%

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

   2. pow2 64.9%

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

   3. sqrt-prod 37.9%

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

   4. sqrt-pow1 39.4%

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

   5. metadata-eval 39.4%

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

   6. pow1 39.4%

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

   7. associate-*r* 39.4%

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

8. Applied egg-rr 39.4%

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

   1. unpow-prod-down 37.9%

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

   2. associate-*r* 37.9%

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

   3. pow2 37.9%

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

   4. pow2 37.9%

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

   5. add-sqr-sqrt 84.8%

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

   6. associate-*r/ 81.9%

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

   7. associate-*r* 81.9%

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

   8. associate-*r* 81.9%

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

   9. associate-*l* 84.6%

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

   10. associate-*r* 84.6%

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

   11. associate-*r* 84.6%

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

10. Applied egg-rr 84.6%

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

    1. associate-*r/ 90.1%

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

    2. associate-*r/ 90.1%

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

12. Applied egg-rr 90.1%

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

13. Final simplification 90.1%

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

Alternative 6: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 9 \cdot 10^{+139}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 9e+139)
   w0
   (+ w0 (* -0.125 (* (pow (* D_m (/ M_m d)) 2.0) (* h (/ w0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 9e+139) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d_m <= 9d+139) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d_m * (m_m / d)) ** 2.0d0) * (h * (w0 / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 9e+139) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if D_m <= 9e+139:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D_m * (M_m / d)), 2.0) * (h * (w0 / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 9e+139)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(h * Float64(w0 / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (D_m <= 9e+139)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D_m * (M_m / d)) ^ 2.0) * (h * (w0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 9e+139], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 8.9999999999999999e139

   1. Initial program 83.5%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in M around 0 74.6%

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

   if 8.9999999999999999e139 < D

   1. Initial program 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in M around 0 25.5%

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

      1. +-commutative 25.5%

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

      2. fma-define 25.5%

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

      3. associate-*r* 25.5%

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

      4. *-commutative 25.5%

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

      5. unpow2 25.5%

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

      6. unpow2 25.5%

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

      7. swap-sqr 48.8%

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

      8. unpow2 48.8%

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

      9. *-commutative 48.8%

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

   7. Simplified 48.8%

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

      1. fma-undefine 48.8%

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

      2. associate-/l* 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. *-commutative 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. unpow2 48.7%

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

   11. Applied egg-rr 48.7%

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

   12. Taylor expanded in M around 0 25.5%

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

       1. associate-*r* 25.5%

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

       2. times-frac 28.6%

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

       3. unpow2 28.6%

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

       4. unpow2 28.6%

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

       5. swap-sqr 51.8%

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

       6. unpow2 51.8%

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

       7. unpow2 51.8%

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

       8. unpow2 51.8%

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

       9. times-frac 61.5%

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

       10. unpow1 61.5%

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

       11. pow-plus 61.5%

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

       12. associate-/l* 61.5%

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

       13. metadata-eval 61.5%

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

       14. associate-/l* 67.5%

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

   14. Simplified 67.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 9 \cdot 10^{+139}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in M around 0 69.2%

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

Reproduce

herbie shell --seed 2024157 
(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))))))