Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 87.1%

Time: 16.3s

Alternatives: 8

Speedup: 1.0×

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \frac{D \cdot 0.5}{d}\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq 2 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* (* M_m (/ (* D 0.5) d)) (sqrt h))))
   (if (<= h 2e-173)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* D M_m) (/ 0.5 d)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ t_0 l))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * ((D * 0.5) / d)) * sqrt(h);
	double tmp;
	if (h <= 2e-173) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D * M_m) * (0.5 / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * ((d * 0.5d0) / d_1)) * sqrt(h)
    if (h <= 2d-173) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((d * m_m) * (0.5d0 / d_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * ((D * 0.5) / d)) * Math.sqrt(h);
	double tmp;
	if (h <= 2e-173) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((D * M_m) * (0.5 / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = (M_m * ((D * 0.5) / d)) * math.sqrt(h)
	tmp = 0
	if h <= 2e-173:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((D * M_m) * (0.5 / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 / l))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(M_m * Float64(Float64(D * 0.5) / d)) * sqrt(h))
	tmp = 0.0
	if (h <= 2e-173)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = (M_m * ((D * 0.5) / d)) * sqrt(h);
	tmp = 0.0;
	if (h <= 2e-173)
		tmp = w0 * sqrt((1.0 - ((h * (((D * M_m) * (0.5 / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 2e-173], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D * M$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 2.0000000000000001e-173

   1. Initial program 84.9%

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

   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. Step-by-step derivation

      1. associate-*r/ 85.6%

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

      2. clear-num 85.6%

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

      3. add-sqr-sqrt 85.6%

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

      4. pow2 85.6%

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

      5. sqrt-pow1 85.6%

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

      6. metadata-eval 85.6%

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

      7. pow1 85.6%

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

      8. *-un-lft-identity 85.6%

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

      9. times-frac 85.6%

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

      10. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r/ 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*r* 85.6%

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

      4. associate-*r/ 86.8%

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

      5. *-commutative 86.8%

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

      6. associate-*r/ 86.8%

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

      7. associate-*r/ 86.8%

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

   8. Simplified 86.8%

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

      1. associate-*l/ 86.8%

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

      2. *-un-lft-identity 86.8%

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

      3. associate-*r* 86.8%

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

   10. Applied egg-rr 86.8%

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

   if 2.0000000000000001e-173 < h

   1. Initial program 76.4%

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

   3. Simplified 76.4%

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

      1. associate-*r/ 86.4%

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

      2. add-sqr-sqrt 86.4%

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

      3. pow2 86.4%

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

      4. sqrt-pow1 86.4%

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

      5. metadata-eval 86.4%

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

      6. pow1 86.4%

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

      7. *-un-lft-identity 86.4%

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

      8. times-frac 86.4%

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

      9. metadata-eval 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. add-sqr-sqrt 86.4%

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

      2. *-un-lft-identity 86.4%

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

      3. times-frac 86.4%

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

      4. sqrt-prod 86.4%

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

      5. sqrt-pow1 76.7%

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

      6. metadata-eval 76.7%

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

      7. pow1 76.7%

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

      8. associate-*r/ 76.7%

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

      9. sqrt-prod 76.7%

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

      10. sqrt-pow1 91.8%

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

      11. metadata-eval 91.8%

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

      12. pow1 91.8%

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

      13. associate-*r/ 91.8%

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

   8. Applied egg-rr 91.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \sqrt{h}\right) \cdot \frac{\left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M\_m \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* D M_m) (* d 2.0)) 2.0) (/ h l)))))
   (if (<= t_0 5e+264)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (/ (* D 0.5) d)) 2.0) l))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((D * M_m) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+264) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * ((D * 0.5) / d)), 2.0) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((d * m_m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+264) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m * ((d * 0.5d0) / d_1)) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((D * M_m) / (d * 2.0)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+264) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * ((D * 0.5) / d)), 2.0) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = 1.0 - (math.pow(((D * M_m) / (d * 2.0)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+264:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M_m * ((D * 0.5) / d)), 2.0) / l))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+264)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(Float64(D * 0.5) / d)) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = 1.0 - ((((D * M_m) / (d * 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+264)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (h * (((M_m * ((D * 0.5) / d)) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+264], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5.00000000000000033e264

   1. Initial program 99.9%

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

   if 5.00000000000000033e264 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

      1. associate-*r/ 58.9%

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

      2. add-sqr-sqrt 58.9%

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

      3. pow2 58.9%

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

      4. sqrt-pow1 58.9%

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

      5. metadata-eval 58.9%

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

      6. pow1 58.9%

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

      7. *-un-lft-identity 58.9%

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

      8. times-frac 58.9%

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

      9. metadata-eval 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. *-un-lft-identity 58.9%

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

      2. associate-*r/ 40.2%

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

      3. add-sqr-sqrt 40.2%

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

      4. pow2 40.2%

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

      5. sqrt-pow1 40.2%

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

      6. metadata-eval 40.2%

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

      7. pow1 40.2%

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

      8. associate-*r/ 40.2%

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

   8. Applied egg-rr 40.2%

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

      1. *-lft-identity 40.2%

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

      2. associate-*r/ 58.9%

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

      3. *-commutative 58.9%

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

      4. associate-*r/ 61.3%

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

      5. *-commutative 61.3%

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

   10. Simplified 61.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(D \cdot M\_m\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}} \end{array} \]

FPCore

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

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow(((D * M_m) * (0.5 / d)), 2.0)) / l)));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - ((h * (((d * m_m) * (0.5d0 / d_1)) ** 2.0d0)) / l)))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((h * Math.pow(((D * M_m) * (0.5 / d)), 2.0)) / l)));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((D * M_m) * (0.5 / d)), 2.0)) / l)))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D * M_m) * Float64(0.5 / d)) ^ 2.0)) / l))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (((D * M_m) * (0.5 / d)) ^ 2.0)) / l)));
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Simplified 81.0%

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

   1. associate-*r/ 85.9%

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

   2. clear-num 85.9%

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

   3. add-sqr-sqrt 85.9%

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

   4. pow2 85.9%

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

   5. sqrt-pow1 85.9%

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

   6. metadata-eval 85.9%

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

   7. pow1 85.9%

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

   8. *-un-lft-identity 85.9%

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

   9. times-frac 85.9%

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

   10. metadata-eval 85.9%

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

6. Applied egg-rr 85.9%

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

   1. associate-/r/ 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*r* 85.9%

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

   4. associate-*r/ 86.7%

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

   5. *-commutative 86.7%

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

   6. associate-*r/ 86.3%

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

   7. associate-*r/ 86.3%

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

8. Simplified 86.3%

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

   1. associate-*l/ 86.3%

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

   2. *-un-lft-identity 86.3%

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

   3. associate-*r* 86.7%

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

10. Applied egg-rr 86.7%

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

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M\_m \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\ell}} \end{array} \]

FPCore

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

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((M_m * ((D * 0.5) / d)), 2.0) / l))));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - (h * (((m_m * ((d * 0.5d0) / d_1)) ** 2.0d0) / l))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * ((D * 0.5) / d)), 2.0) / l))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M_m * ((D * 0.5) / d)), 2.0) / l))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(Float64(D * 0.5) / d)) ^ 2.0) / l)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M_m * ((D * 0.5) / d)) ^ 2.0) / l))));
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Simplified 81.0%

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

   1. associate-*r/ 85.9%

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

   2. add-sqr-sqrt 85.9%

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

   3. pow2 85.9%

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

   4. sqrt-pow1 85.9%

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

   5. metadata-eval 85.9%

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

   6. pow1 85.9%

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

   7. *-un-lft-identity 85.9%

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

   8. times-frac 85.9%

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

   9. metadata-eval 85.9%

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

6. Applied egg-rr 85.9%

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

   1. *-un-lft-identity 85.9%

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

   2. associate-*r/ 81.0%

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

   3. add-sqr-sqrt 81.0%

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

   4. pow2 81.0%

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

   5. sqrt-pow1 81.0%

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

   6. metadata-eval 81.0%

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

   7. pow1 81.0%

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

   8. associate-*r/ 81.0%

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

8. Applied egg-rr 81.0%

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

   1. *-lft-identity 81.0%

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

   2. associate-*r/ 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*r/ 85.9%

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

   5. *-commutative 85.9%

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

10. Simplified 85.9%

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

Alternative 5: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(D \cdot M\_m\right)}^{2} \cdot \left(\frac{1}{\ell} \cdot \left(\frac{1}{d} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= D 2.6e+14)
   w0
   (*
    w0
    (+
     1.0
     (* (* (pow (* D M_m) 2.0) (* (/ 1.0 l) (* (/ 1.0 d) (/ h d)))) -0.125)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.6e+14) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((pow((D * M_m), 2.0) * ((1.0 / l) * ((1.0 / d) * (h / d)))) * -0.125));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.6d+14) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((((d * m_m) ** 2.0d0) * ((1.0d0 / l) * ((1.0d0 / d_1) * (h / d_1)))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.6e+14) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((Math.pow((D * M_m), 2.0) * ((1.0 / l) * ((1.0 / d) * (h / d)))) * -0.125));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if D <= 2.6e+14:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((math.pow((D * M_m), 2.0) * ((1.0 / l) * ((1.0 / d) * (h / d)))) * -0.125))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (D <= 2.6e+14)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64((Float64(D * M_m) ^ 2.0) * Float64(Float64(1.0 / l) * Float64(Float64(1.0 / d) * Float64(h / d)))) * -0.125)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (D <= 2.6e+14)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((((D * M_m) ^ 2.0) * ((1.0 / l) * ((1.0 / d) * (h / d)))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[D, 2.6e+14], w0, N[(w0 * N[(1.0 + N[(N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(1.0 / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.6e14

   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 85.7%

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

   5. Taylor expanded in M around 0 72.4%

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

   if 2.6e14 < D

   1. Initial program 67.8%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in M around 0 34.4%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 34.4%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.125}\right) \]

      2. associate-/l* 34.4%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]

      3. associate-/l* 36.2%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) \cdot -0.125\right) \]

   7. Simplified 36.2%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot -0.125\right)} \]

   8. Taylor expanded in D around 0 34.4%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. associate-*r* 34.5%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.125\right) \]

      2. *-commutative 34.5%

         \[\leadsto w0 \cdot \left(1 + \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\ell \cdot {d}^{2}}} \cdot -0.125\right) \]

      3. unpow2 34.5%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      4. unpow2 34.5%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      5. swap-sqr 57.6%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      6. unpow2 57.6%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      7. associate-*r/ 57.5%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell \cdot {d}^{2}}\right)} \cdot -0.125\right) \]

      8. *-commutative 57.5%

         \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\color{blue}{{d}^{2} \cdot \ell}}\right) \cdot -0.125\right) \]

   10. Simplified 57.5%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. *-un-lft-identity 57.5%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \frac{\color{blue}{1 \cdot h}}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right) \]

       2. *-commutative 57.5%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \frac{1 \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right) \cdot -0.125\right) \]

       3. times-frac 57.5%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{1}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\right) \cdot -0.125\right) \]

   12. Applied egg-rr 57.5%

       \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{1}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\right) \cdot -0.125\right) \]
   13. Step-by-step derivation

       1. *-un-lft-identity 57.5%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \left(\frac{1}{\ell} \cdot \frac{\color{blue}{1 \cdot h}}{{d}^{2}}\right)\right) \cdot -0.125\right) \]

       2. unpow2 57.5%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \left(\frac{1}{\ell} \cdot \frac{1 \cdot h}{\color{blue}{d \cdot d}}\right)\right) \cdot -0.125\right) \]

       3. times-frac 62.7%

          \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\left(\frac{1}{d} \cdot \frac{h}{d}\right)}\right)\right) \cdot -0.125\right) \]

   14. Applied egg-rr 62.7%

       \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\left(\frac{1}{d} \cdot \frac{h}{d}\right)}\right)\right) \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-143}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\left(D \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{h}{\ell \cdot {d}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (/ h l) -1e-143)
   (*
    w0
    (+ 1.0 (* -0.125 (* (* (* D M_m) (* D M_m)) (/ h (* l (pow d 2.0)))))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-143) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h / (l * pow(d, 2.0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-1d-143)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * m_m) * (d * m_m)) * (h / (l * (d_1 ** 2.0d0))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-143) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h / (l * Math.pow(d, 2.0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-143:
		tmp = w0 * (1.0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h / (l * math.pow(d, 2.0))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-143)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * M_m) * Float64(D * M_m)) * Float64(h / Float64(l * (d ^ 2.0)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-143)
		tmp = w0 * (1.0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h / (l * (d ^ 2.0))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-143], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.9999999999999995e-144

   1. Initial program 73.9%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in M around 0 42.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 42.1%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.125}\right) \]

      2. associate-/l* 43.0%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]

      3. associate-/l* 44.7%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) \cdot -0.125\right) \]

   7. Simplified 44.7%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot -0.125\right)} \]

   8. Taylor expanded in D around 0 42.1%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. associate-*r* 44.7%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.125\right) \]

      2. *-commutative 44.7%

         \[\leadsto w0 \cdot \left(1 + \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\ell \cdot {d}^{2}}} \cdot -0.125\right) \]

      3. unpow2 44.7%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      4. unpow2 44.7%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      5. swap-sqr 55.6%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      6. unpow2 55.6%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      7. associate-*r/ 56.6%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell \cdot {d}^{2}}\right)} \cdot -0.125\right) \]

      8. *-commutative 56.6%

         \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\color{blue}{{d}^{2} \cdot \ell}}\right) \cdot -0.125\right) \]

   10. Simplified 56.6%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]
   11. Step-by-step derivation

       1. unpow2 56.6%

          \[\leadsto w0 \cdot \left(1 + \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right) \]

   12. Applied egg-rr 56.6%

       \[\leadsto w0 \cdot \left(1 + \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right) \]

   if -9.9999999999999995e-144 < (/.f64 h l)

   1. Initial program 88.1%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in M around 0 89.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-143}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\ell \cdot {d}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1200000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{h \cdot w0}{\ell \cdot {d}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 1200000000.0)
   w0
   (* -0.125 (* (* (* D M_m) (* D M_m)) (/ (* h w0) (* l (pow d 2.0)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1200000000.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M_m) * (D * M_m)) * ((h * w0) / (l * pow(d, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 1200000000.0d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * m_m) * (d * m_m)) * ((h * w0) / (l * (d_1 ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1200000000.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M_m) * (D * M_m)) * ((h * w0) / (l * Math.pow(d, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 1200000000.0:
		tmp = w0
	else:
		tmp = -0.125 * (((D * M_m) * (D * M_m)) * ((h * w0) / (l * math.pow(d, 2.0))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 1200000000.0)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * M_m) * Float64(D * M_m)) * Float64(Float64(h * w0) / Float64(l * (d ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 1200000000.0)
		tmp = w0;
	else
		tmp = -0.125 * (((D * M_m) * (D * M_m)) * ((h * w0) / (l * (d ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 1200000000.0], w0, N[(-0.125 * N[(N[(N[(D * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.2e9

   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 84.7%

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

   5. Taylor expanded in M around 0 73.0%

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

   if 1.2e9 < M

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in M around 0 46.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.125}\right) \]

      2. associate-/l* 46.1%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]

      3. associate-/l* 46.0%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) \cdot -0.125\right) \]

   7. Simplified 46.0%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left({D}^{2} \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) \cdot -0.125\right)} \]

   8. Taylor expanded in D around 0 46.1%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. associate-*r* 44.2%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.125\right) \]

      2. *-commutative 44.2%

         \[\leadsto w0 \cdot \left(1 + \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\ell \cdot {d}^{2}}} \cdot -0.125\right) \]

      3. unpow2 44.2%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      4. unpow2 44.2%

         \[\leadsto w0 \cdot \left(1 + \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      5. swap-sqr 61.8%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      6. unpow2 61.8%

         \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\ell \cdot {d}^{2}} \cdot -0.125\right) \]

      7. associate-*r/ 61.7%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell \cdot {d}^{2}}\right)} \cdot -0.125\right) \]

      8. *-commutative 61.7%

         \[\leadsto w0 \cdot \left(1 + \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\color{blue}{{d}^{2} \cdot \ell}}\right) \cdot -0.125\right) \]

   10. Simplified 61.7%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]

   11. Taylor expanded in D around inf 21.3%

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

       1. associate-*r* 21.3%

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

       2. unpow2 21.3%

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

       3. unpow2 21.3%

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

       4. swap-sqr 22.0%

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

       5. unpow2 22.0%

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

       6. associate-*r/ 21.9%

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

   13. Simplified 21.9%

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

       1. unpow2 61.7%

          \[\leadsto w0 \cdot \left(1 + \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right) \]

   15. Applied egg-rr 21.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1200000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h \cdot w0}{\ell \cdot {d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Simplified 81.0%

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

5. Taylor expanded in M around 0 68.2%

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

Reproduce

herbie shell --seed 2024107 
(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))))))