Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 87.3%

Time: 16.7s

Alternatives: 6

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: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{2} \cdot \frac{M\_m}{d}\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq 2 \cdot 10^{-213}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot M\_m}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0}{\ell}}\\ \end{array} \end{array} \]

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 (* (* (/ D 2.0) (/ M_m d)) (sqrt h))))
   (if (<= h 2e-213)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* D M_m) (* 2.0 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 = ((D / 2.0) * (M_m / d)) * sqrt(h);
	double tmp;
	if (h <= 2e-213) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D * M_m) / (2.0 * 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 = ((d / 2.0d0) * (m_m / d_1)) * sqrt(h)
    if (h <= 2d-213) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((d * m_m) / (2.0d0 * 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 = ((D / 2.0) * (M_m / d)) * Math.sqrt(h);
	double tmp;
	if (h <= 2e-213) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((D * M_m) / (2.0 * 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 = ((D / 2.0) * (M_m / d)) * math.sqrt(h)
	tmp = 0
	if h <= 2e-213:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((D * M_m) / (2.0 * 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(Float64(D / 2.0) * Float64(M_m / d)) * sqrt(h))
	tmp = 0.0
	if (h <= 2e-213)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D * M_m) / Float64(2.0 * 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 = ((D / 2.0) * (M_m / d)) * sqrt(h);
	tmp = 0.0;
	if (h <= 2e-213)
		tmp = w0 * sqrt((1.0 - ((h * (((D * M_m) / (2.0 * 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[(N[(D / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 2e-213], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(2.0 * 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 < 1.9999999999999999e-213

   1. Initial program 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 86.4%

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

      2. unpow2 86.4%

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

      3. associate-*l* 86.4%

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

      4. associate-/r* 86.4%

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

      5. associate-*r/ 85.8%

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

      6. associate-*l/ 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-/r* 85.8%

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

      9. associate-*r/ 87.1%

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

      10. associate-*l/ 87.7%

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

      11. *-commutative 87.7%

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

      12. associate-*l* 86.4%

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

      13. unpow2 86.4%

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

   6. Applied egg-rr 86.5%

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

   if 1.9999999999999999e-213 < h

   1. Initial program 85.4%

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

   3. Simplified 84.4%

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

      1. associate-*r/ 88.8%

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

      2. unpow2 88.8%

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

      3. associate-*l* 90.4%

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

      4. associate-/r* 90.4%

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

      5. associate-*r/ 90.4%

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

      6. associate-*l/ 90.4%

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

      7. *-commutative 90.4%

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

      8. associate-/r* 90.4%

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

      9. associate-*r/ 91.4%

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

      10. associate-*l/ 91.4%

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

      11. *-commutative 91.4%

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

      12. associate-*l* 89.8%

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

      13. unpow2 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. add-sqr-sqrt 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 89.8%

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

      4. *-commutative 89.8%

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

      5. sqrt-prod 89.8%

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

      6. sqrt-pow1 74.8%

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

      7. metadata-eval 74.8%

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

      8. pow1 74.8%

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

      9. times-frac 73.8%

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

   8. Applied egg-rr 94.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2 \cdot 10^{-213}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* D M_m) (* 2.0 d)) 2.0) (/ h l)) 1e-15)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D (/ (/ M_m 2.0) 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 ((pow(((D * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 1e-15) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D * ((M_m / 2.0) / 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 (((((d * m_m) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= 1d-15) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d * ((m_m / 2.0d0) / 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 ((Math.pow(((D * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 1e-15) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D * ((M_m / 2.0) / 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 (math.pow(((D * M_m) / (2.0 * d)), 2.0) * (h / l)) <= 1e-15:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D * ((M_m / 2.0) / 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((Float64(Float64(D * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 1e-15)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D * Float64(Float64(M_m / 2.0) / 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 (((((D * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= 1e-15)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * ((M_m / 2.0) / 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[(N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-15], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1.0000000000000001e-15

   1. Initial program 89.1%

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

   3. Simplified 88.7%

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

   if 1.0000000000000001e-15 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in D around 0 66.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 1.0× speedup

Math

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

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) (* 2.0 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) / (2.0 * 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) / (2.0d0 * 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) / (2.0 * 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) / (2.0 * 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(2.0 * 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) / (2.0 * 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[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

3. Simplified 82.1%

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

   1. associate-*r/ 87.4%

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

   2. unpow2 87.4%

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

   3. associate-*l* 88.0%

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

   4. associate-/r* 88.0%

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

   5. associate-*r/ 87.6%

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

   6. associate-*l/ 87.6%

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

   7. *-commutative 87.6%

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

   8. associate-/r* 87.6%

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

   9. associate-*r/ 88.8%

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

   10. associate-*l/ 89.2%

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

   11. *-commutative 89.2%

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

   12. associate-*l* 87.8%

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

   13. unpow2 87.8%

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

6. Applied egg-rr 87.8%

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

7. Final simplification 87.8%

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

Alternative 4: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{+100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot M\_m\right)}^{2} \cdot \frac{h \cdot \frac{w0}{d}}{d \cdot \ell}\right)\\ \end{array} \end{array} \]

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 4e+100)
   w0
   (+ w0 (* -0.125 (* (pow (* D M_m) 2.0) (/ (* h (/ w0 d)) (* d 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 tmp;
	if (D <= 4e+100) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D * M_m), 2.0) * ((h * (w0 / d)) / (d * 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) :: tmp
    if (d <= 4d+100) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * m_m) ** 2.0d0) * ((h * (w0 / d_1)) / (d_1 * 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 tmp;
	if (D <= 4e+100) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D * M_m), 2.0) * ((h * (w0 / d)) / (d * 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):
	tmp = 0
	if D <= 4e+100:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D * M_m), 2.0) * ((h * (w0 / d)) / (d * 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)
	tmp = 0.0
	if (D <= 4e+100)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D * M_m) ^ 2.0) * Float64(Float64(h * Float64(w0 / d)) / Float64(d * 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)
	tmp = 0.0;
	if (D <= 4e+100)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * M_m) ^ 2.0) * ((h * (w0 / d)) / (d * 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_] := If[LessEqual[D, 4e+100], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * N[(w0 / d), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.00000000000000006e100

   1. Initial program 86.1%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in D around 0 73.5%

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

   if 4.00000000000000006e100 < D

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in D around 0 32.5%

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

      1. associate-/l* 32.5%

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

      2. associate-/l* 32.1%

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

   7. Simplified 32.1%

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

   8. Taylor expanded in D around 0 32.5%

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

      1. associate-*r* 32.6%

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

      2. unpow2 32.6%

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

      3. unpow2 32.6%

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

      4. swap-sqr 54.0%

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

      5. unpow2 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r/ 53.7%

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

      8. associate-*r/ 51.2%

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

      9. associate-/r* 51.0%

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

   10. Simplified 51.0%

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

       1. div-inv 51.0%

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

       2. unpow2 51.0%

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

       3. times-frac 51.4%

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

   12. Applied egg-rr 51.4%

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

       1. associate-*r* 51.4%

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

       2. associate-/l/ 51.3%

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

       3. un-div-inv 51.3%

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

   14. Applied egg-rr 51.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{+100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h \cdot \frac{w0}{d}}{d \cdot \ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)\right) \cdot \left(h \cdot \left(\frac{w0}{d} \cdot \frac{\frac{1}{\ell}}{d}\right)\right)\right)\\ \end{array} \end{array} \]

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 4.4e+100)
   w0
   (+
    w0
    (*
     -0.125
     (* (* (* D M_m) (* D M_m)) (* h (* (/ w0 d) (/ (/ 1.0 l) d))))))))

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 <= 4.4e+100) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h * ((w0 / d) * ((1.0 / l) / d)))));
	}
	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 <= 4.4d+100) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * m_m) * (d * m_m)) * (h * ((w0 / d_1) * ((1.0d0 / l) / d_1)))))
    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 <= 4.4e+100) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h * ((w0 / d) * ((1.0 / l) / d)))));
	}
	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 <= 4.4e+100:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h * ((w0 / d) * ((1.0 / l) / d)))))
	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 <= 4.4e+100)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D * M_m) * Float64(D * M_m)) * Float64(h * Float64(Float64(w0 / d) * Float64(Float64(1.0 / l) / d))))));
	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 <= 4.4e+100)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * M_m) * (D * M_m)) * (h * ((w0 / d) * ((1.0 / l) / d)))));
	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, 4.4e+100], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(w0 / d), $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.4000000000000001e100

   1. Initial program 86.1%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in D around 0 73.5%

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

   if 4.4000000000000001e100 < D

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in D around 0 32.5%

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

      1. associate-/l* 32.5%

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

      2. associate-/l* 32.1%

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

   7. Simplified 32.1%

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

   8. Taylor expanded in D around 0 32.5%

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

      1. associate-*r* 32.6%

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

      2. unpow2 32.6%

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

      3. unpow2 32.6%

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

      4. swap-sqr 54.0%

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

      5. unpow2 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r/ 53.7%

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

      8. associate-*r/ 51.2%

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

      9. associate-/r* 51.0%

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

   10. Simplified 51.0%

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

       1. div-inv 51.0%

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

       2. unpow2 51.0%

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

       3. times-frac 51.4%

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

   12. Applied egg-rr 51.4%

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

       1. unpow2 51.4%

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

   14. Applied egg-rr 51.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(h \cdot \left(\frac{w0}{d} \cdot \frac{\frac{1}{\ell}}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% 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 82.5%

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

3. Simplified 82.1%

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

5. Taylor expanded in D around 0 65.9%

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

6. Final simplification 65.9%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(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))))))