Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 88.1%

Time: 19.3s

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: 81.5% 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: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M\_m}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(\frac{M\_m}{d} \cdot \frac{D}{2}\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= h -2e-308)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* D (/ M_m (* 2.0 d))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ (pow (* (* (/ M_m d) (/ D 2.0)) (sqrt h)) 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 tmp;
	if (h <= -2e-308) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D * (M_m / (2.0 * d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((((M_m / d) * (D / 2.0)) * sqrt(h)), 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) :: tmp
    if (h <= (-2d-308)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d * (m_m / (2.0d0 * d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((((m_m / d_1) * (d / 2.0d0)) * sqrt(h)) ** 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 tmp;
	if (h <= -2e-308) {
		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 - (Math.pow((((M_m / d) * (D / 2.0)) * Math.sqrt(h)), 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):
	tmp = 0
	if h <= -2e-308:
		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 - (math.pow((((M_m / d) * (D / 2.0)) * math.sqrt(h)), 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)
	tmp = 0.0
	if (h <= -2e-308)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D * Float64(M_m / Float64(2.0 * d))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(Float64(M_m / d) * Float64(D / 2.0)) * sqrt(h)) ^ 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)
	tmp = 0.0;
	if (h <= -2e-308)
		tmp = w0 * sqrt((1.0 - ((h * ((D * (M_m / (2.0 * d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((((M_m / d) * (D / 2.0)) * sqrt(h)) ^ 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_] := If[LessEqual[h, -2e-308], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.9999999999999998e-308

   1. Initial program 80.4%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 85.2%

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

      2. add-sqr-sqrt 85.2%

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

      3. pow2 85.2%

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

      4. unpow2 85.2%

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

      5. sqrt-prod 57.1%

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

      6. add-sqr-sqrt 85.2%

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

      7. div-inv 85.2%

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

      8. frac-times 85.2%

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

      9. *-commutative 85.2%

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

      10. *-un-lft-identity 85.2%

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

   6. Applied egg-rr 85.2%

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

   if -1.9999999999999998e-308 < h

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

      1. associate-*r/ 83.3%

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

      2. add-sqr-sqrt 83.3%

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

      3. pow2 83.3%

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

      4. unpow2 83.3%

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

      5. sqrt-prod 51.1%

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

      6. add-sqr-sqrt 83.3%

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

      7. div-inv 83.3%

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

      8. frac-times 83.3%

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

      9. *-commutative 83.3%

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

      10. *-un-lft-identity 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. add-sqr-sqrt 83.3%

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

      2. pow2 83.3%

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

      3. sqrt-prod 83.3%

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

      4. sqrt-pow1 87.9%

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

      5. metadata-eval 87.9%

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

      6. pow1 87.9%

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

   8. Applied egg-rr 87.9%

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

      1. associate-*r/ 87.1%

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

      2. *-commutative 87.1%

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

      3. *-commutative 87.1%

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

      4. times-frac 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2 \cdot d}{M\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) 5e-8)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (/ (* 2.0 d) M_m)) 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)) <= 5e-8) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / ((2.0 * d) / M_m)), 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)) <= 5d-8) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / ((2.0d0 * d_1) / m_m)) ** 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)) <= 5e-8) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / ((2.0 * d) / M_m)), 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)) <= 5e-8:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / ((2.0 * d) / M_m)), 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)) <= 5e-8)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(Float64(2.0 * d) / M_m)) ^ 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)) <= 5e-8)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / ((2.0 * d) / M_m)) ^ 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], 5e-8], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(N[(2.0 * d), $MachinePrecision] / M$95$m), $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)) < 4.9999999999999998e-8

   1. Initial program 87.5%

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

   3. Simplified 87.5%

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

      1. clear-num 87.5%

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

      2. un-div-inv 87.5%

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

      3. div-inv 87.5%

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

      4. clear-num 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. *-un-lft-identity 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. *-lft-identity 87.5%

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

      2. associate-*r/ 87.5%

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

      3. *-commutative 87.5%

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

   10. Simplified 87.5%

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

   if 4.9999999999999998e-8 < (*.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 3.8%

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

   5. Taylor expanded in D around 0 63.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2 \cdot d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) -4e-311)
   (* 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 ((h / l) <= -4e-311) {
		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 ((h / l) <= (-4d-311)) 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 ((h / l) <= -4e-311) {
		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 (h / l) <= -4e-311:
		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(h / l) <= -4e-311)
		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 ((h / l) <= -4e-311)
		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[(h / l), $MachinePrecision], -4e-311], 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 h l) < -3.99999999999979e-311

   1. Initial program 75.7%

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

   3. Simplified 75.7%

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

   if -3.99999999999979e-311 < (/.f64 h l)

   1. Initial program 82.7%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in D around 0 91.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) -4e-311)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (* d (/ 2.0 M_m))) 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) <= -4e-311) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / (d * (2.0 / M_m))), 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) <= (-4d-311)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / (d_1 * (2.0d0 / m_m))) ** 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) <= -4e-311) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / (d * (2.0 / M_m))), 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) <= -4e-311:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / (d * (2.0 / M_m))), 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) <= -4e-311)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(d * Float64(2.0 / M_m))) ^ 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) <= -4e-311)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / (d * (2.0 / M_m))) ^ 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], -4e-311], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -3.99999999999979e-311

   1. Initial program 75.7%

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

   3. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. un-div-inv 75.7%

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

      3. div-inv 75.6%

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

      4. clear-num 75.6%

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

   6. Applied egg-rr 75.6%

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

   if -3.99999999999979e-311 < (/.f64 h l)

   1. Initial program 82.7%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in D around 0 91.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.0× speedup

Math

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

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(D * Float64(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[(D * N[(M$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 79.0%

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

   1. associate-*r/ 84.3%

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

   2. add-sqr-sqrt 84.3%

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

   3. pow2 84.3%

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

   4. unpow2 84.3%

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

   5. sqrt-prod 54.2%

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

   6. add-sqr-sqrt 84.3%

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

   7. div-inv 84.3%

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

   8. frac-times 84.3%

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

   9. *-commutative 84.3%

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

   10. *-un-lft-identity 84.3%

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

6. Applied egg-rr 84.3%

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

7. Final simplification 84.3%

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

Alternative 6: 73.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M\_m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \end{array} \end{array} \]

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 M_m) 5e-8)
   w0
   (+ w0 (* -0.125 (* (pow (* D (/ M_m d)) 2.0) (/ (* h w0) 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 * M_m) <= 5e-8) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D * (M_m / d)), 2.0) * ((h * w0) / 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 * m_m) <= 5d-8) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * (m_m / d_1)) ** 2.0d0) * ((h * w0) / 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 * M_m) <= 5e-8) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D * (M_m / d)), 2.0) * ((h * w0) / 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 * M_m) <= 5e-8:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D * (M_m / d)), 2.0) * ((h * w0) / 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 (Float64(D * M_m) <= 5e-8)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(Float64(h * w0) / 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 * M_m) <= 5e-8)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * (M_m / d)) ^ 2.0) * ((h * w0) / 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[N[(D * M$95$m), $MachinePrecision], 5e-8], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4.9999999999999998e-8

   1. Initial program 80.7%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in D around 0 74.2%

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

   if 4.9999999999999998e-8 < (*.f64 M D)

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in D around 0 39.9%

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

      1. associate-/l* 39.9%

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

      2. times-frac 40.0%

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

   7. Simplified 40.0%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-div 40.0%

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

      3. sqrt-pow1 24.1%

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

      4. metadata-eval 24.1%

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

      5. pow1 24.1%

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

      6. sqrt-pow1 26.7%

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

      7. metadata-eval 26.7%

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

      8. pow1 26.7%

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

      9. sqrt-div 26.7%

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

      10. sqrt-pow1 40.6%

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

      11. metadata-eval 40.6%

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

      12. pow1 40.6%

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

      13. sqrt-pow1 51.5%

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

      14. metadata-eval 51.5%

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

      15. pow1 51.5%

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

   9. Applied egg-rr 51.5%

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

       1. pow1 51.5%

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

       2. associate-*r* 51.5%

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

       3. pow2 51.5%

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

       4. pow-prod-down 62.9%

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

       5. associate-/l* 61.5%

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

   11. Applied egg-rr 61.5%

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

       1. unpow1 61.5%

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

   13. Simplified 61.5%

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

   14. Taylor expanded in h around 0 62.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D (/ M_m d))))
   (if (<= d 1.15e+41) (+ w0 (* -0.125 (* (* t_0 t_0) (* h (/ w0 l))))) 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 t_0 = D * (M_m / d);
	double tmp;
	if (d <= 1.15e+41) {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = d * (m_m / d_1)
    if (d_1 <= 1.15d+41) then
        tmp = w0 + ((-0.125d0) * ((t_0 * t_0) * (h * (w0 / l))))
    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 t_0 = D * (M_m / d);
	double tmp;
	if (d <= 1.15e+41) {
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	} 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):
	t_0 = D * (M_m / d)
	tmp = 0
	if d <= 1.15e+41:
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))))
	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)
	t_0 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (d <= 1.15e+41)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(h * Float64(w0 / l)))));
	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)
	t_0 = D * (M_m / d);
	tmp = 0.0;
	if (d <= 1.15e+41)
		tmp = w0 + (-0.125 * ((t_0 * t_0) * (h * (w0 / l))));
	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_] := Block[{t$95$0 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1.15e+41], N[(w0 + N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < 1.1499999999999999e41

   1. Initial program 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in D around 0 45.2%

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

      1. associate-/l* 45.2%

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

      2. times-frac 44.3%

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

   7. Simplified 44.3%

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

      1. add-sqr-sqrt 44.3%

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

      2. sqrt-div 44.3%

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

      3. sqrt-pow1 34.6%

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

      4. metadata-eval 34.6%

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

      5. pow1 34.6%

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

      6. sqrt-pow1 36.6%

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

      7. metadata-eval 36.6%

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

      8. pow1 36.6%

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

      9. sqrt-div 36.6%

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

      10. sqrt-pow1 39.5%

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

      11. metadata-eval 39.5%

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

      12. pow1 39.5%

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

      13. sqrt-pow1 53.3%

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

      14. metadata-eval 53.3%

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

      15. pow1 53.3%

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

   9. Applied egg-rr 53.3%

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

       1. pow1 53.3%

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

       2. associate-*r* 53.8%

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

       3. pow2 53.8%

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

       4. pow-prod-down 66.8%

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

       5. associate-/l* 67.4%

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

   11. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

   13. Simplified 67.4%

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

       1. unpow2 67.4%

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

   15. Applied egg-rr 67.4%

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

   if 1.1499999999999999e41 < d

   1. Initial program 78.2%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in D around 0 84.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.8% 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 78.6%

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

3. Simplified 79.0%

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

5. Taylor expanded in D around 0 65.8%

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

6. Final simplification 65.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))))))