Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 90.4%

Time: 12.4s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{D \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}}{\frac{d}{M}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 2e+136)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M D) d)) 2.0) h) l))))
   (* w0 (/ (* D (sqrt (/ (* h -0.25) l))) (/ d M)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 2e+136) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M * D) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * ((D * sqrt(((h * -0.25) / l))) / (d / M));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 2d+136) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m * d) / d_1)) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * ((d * sqrt(((h * (-0.25d0)) / l))) / (d_1 / m))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 2e+136) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * ((D * Math.sqrt(((h * -0.25) / l))) / (d / M));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 2e+136:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)))
	else:
		tmp = w0 * ((D * math.sqrt(((h * -0.25) / l))) / (d / M))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 2e+136)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * Float64(Float64(D * sqrt(Float64(Float64(h * -0.25) / l))) / Float64(d / M)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 2e+136)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M * D) / d)) ^ 2.0) * h) / l)));
	else
		tmp = w0 * ((D * sqrt(((h * -0.25) / l))) / (d / M));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e+136], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(D * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 2.00000000000000012e136

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

      1. associate-*r/ 90.6%

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

      2. frac-times 90.6%

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

      3. *-commutative 90.6%

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

      4. unpow2 90.6%

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

      5. unpow2 90.6%

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

      6. *-un-lft-identity 90.6%

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

      7. times-frac 90.6%

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

      8. metadata-eval 90.6%

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

      9. *-commutative 90.6%

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

   5. Applied egg-rr 90.6%

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

   if 2.00000000000000012e136 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 58.7%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in D around inf 54.1%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 54.1%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.25}} \]

      2. associate-*r* 54.1%

         \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.25} \]

      3. times-frac 49.6%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot -0.25} \]

      4. unpow2 49.6%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      5. unpow2 49.6%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      6. swap-sqr 58.5%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      7. unpow2 58.5%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

   6. Simplified 58.5%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25}} \]
   7. Step-by-step derivation

      1. associate-*l* 58.5%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \left(\frac{h}{\ell} \cdot -0.25\right)}} \]

      2. sqrt-prod 58.5%

         \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

      3. sqrt-div 62.7%

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{\sqrt{{\left(D \cdot M\right)}^{2}}}{\sqrt{{d}^{2}}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      4. unpow2 62.7%

         \[\leadsto w0 \cdot \left(\frac{\sqrt{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      5. sqrt-prod 27.2%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{\sqrt{D \cdot M} \cdot \sqrt{D \cdot M}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      6. add-sqr-sqrt 27.3%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{D \cdot M}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      7. unpow2 27.3%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\sqrt{\color{blue}{d \cdot d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      8. sqrt-prod 31.3%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      9. add-sqr-sqrt 71.1%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{d}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      10. associate-/l* 72.2%

          \[\leadsto w0 \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

   8. Applied egg-rr 72.2%

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 72.1%

         \[\leadsto w0 \cdot \color{blue}{\frac{D \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}}{\frac{d}{M}}} \]

      2. associate-*l/ 72.1%

         \[\leadsto w0 \cdot \frac{D \cdot \sqrt{\color{blue}{\frac{h \cdot -0.25}{\ell}}}}{\frac{d}{M}} \]

   10. Applied egg-rr 72.1%

       \[\leadsto w0 \cdot \color{blue}{\frac{D \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}}{\frac{d}{M}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{D \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}}{\frac{d}{M}}\\ \end{array} \]

Alternative 2: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+15)
   (* w0 (* (/ D (/ d M)) (sqrt (* -0.25 (/ h l)))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * ((D / (d / M)) * sqrt((-0.25 * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+15)) then
        tmp = w0 * ((d / (d_1 / m)) * sqrt(((-0.25d0) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * ((D / (d / M)) * Math.sqrt((-0.25 * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+15:
		tmp = w0 * ((D / (d / M)) * math.sqrt((-0.25 * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = Float64(w0 * Float64(Float64(D / Float64(d / M)) * sqrt(Float64(-0.25 * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+15)
		tmp = w0 * ((D / (d / M)) * sqrt((-0.25 * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(w0 * N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2e15

   1. Initial program 69.8%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in D around inf 39.0%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 39.0%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.25}} \]

      2. associate-*r* 39.0%

         \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.25} \]

      3. times-frac 42.8%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot -0.25} \]

      4. unpow2 42.8%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      5. unpow2 42.8%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      6. swap-sqr 51.7%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      7. unpow2 51.7%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

   6. Simplified 51.7%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25}} \]
   7. Step-by-step derivation

      1. associate-*l* 51.7%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \left(\frac{h}{\ell} \cdot -0.25\right)}} \]

      2. sqrt-prod 51.7%

         \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

      3. sqrt-div 52.9%

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{\sqrt{{\left(D \cdot M\right)}^{2}}}{\sqrt{{d}^{2}}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      4. unpow2 52.9%

         \[\leadsto w0 \cdot \left(\frac{\sqrt{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      5. sqrt-prod 30.1%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{\sqrt{D \cdot M} \cdot \sqrt{D \cdot M}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      6. add-sqr-sqrt 30.4%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{D \cdot M}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      7. unpow2 30.4%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\sqrt{\color{blue}{d \cdot d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      8. sqrt-prod 12.5%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      9. add-sqr-sqrt 37.2%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{d}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      10. associate-/l* 37.5%

          \[\leadsto w0 \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

   8. Applied egg-rr 37.5%

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

   if -2e15 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in D around 0 95.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \left(\frac{D}{\frac{d}{M}} \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \frac{D}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.75e-9) w0 (* w0 (* M (* (sqrt (/ (* h -0.25) l)) (/ D d))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.75e-9) {
		tmp = w0;
	} else {
		tmp = w0 * (M * (sqrt(((h * -0.25) / l)) * (D / d)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 1.75d-9) then
        tmp = w0
    else
        tmp = w0 * (m * (sqrt(((h * (-0.25d0)) / l)) * (d / d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.75e-9) {
		tmp = w0;
	} else {
		tmp = w0 * (M * (Math.sqrt(((h * -0.25) / l)) * (D / d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.75e-9:
		tmp = w0
	else:
		tmp = w0 * (M * (math.sqrt(((h * -0.25) / l)) * (D / d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.75e-9)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(M * Float64(sqrt(Float64(Float64(h * -0.25) / l)) * Float64(D / d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.75e-9)
		tmp = w0;
	else
		tmp = w0 * (M * (sqrt(((h * -0.25) / l)) * (D / d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.75e-9], w0, N[(w0 * N[(M * N[(N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.75e-9

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in D around 0 73.6%

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

   if 1.75e-9 < M

   1. Initial program 71.2%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in D around inf 26.9%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 26.9%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.25}} \]

      2. associate-*r* 26.9%

         \[\leadsto w0 \cdot \sqrt{\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot -0.25} \]

      3. times-frac 25.3%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot -0.25} \]

      4. unpow2 25.3%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      5. unpow2 25.3%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      6. swap-sqr 28.1%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

      7. unpow2 28.1%

         \[\leadsto w0 \cdot \sqrt{\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25} \]

   6. Simplified 28.1%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot -0.25}} \]
   7. Step-by-step derivation

      1. associate-*l* 28.1%

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \left(\frac{h}{\ell} \cdot -0.25\right)}} \]

      2. sqrt-prod 27.8%

         \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

      3. sqrt-div 29.7%

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{\sqrt{{\left(D \cdot M\right)}^{2}}}{\sqrt{{d}^{2}}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      4. unpow2 29.7%

         \[\leadsto w0 \cdot \left(\frac{\sqrt{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      5. sqrt-prod 17.4%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{\sqrt{D \cdot M} \cdot \sqrt{D \cdot M}}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      6. add-sqr-sqrt 18.0%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{D \cdot M}}{\sqrt{{d}^{2}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      7. unpow2 18.0%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\sqrt{\color{blue}{d \cdot d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      8. sqrt-prod 10.0%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      9. add-sqr-sqrt 21.6%

         \[\leadsto w0 \cdot \left(\frac{D \cdot M}{\color{blue}{d}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

      10. associate-/l* 22.0%

          \[\leadsto w0 \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]

   8. Applied egg-rr 22.0%

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{\frac{d}{M}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

   9. Taylor expanded in D around 0 21.6%

      \[\leadsto w0 \cdot \left(\color{blue}{\frac{D \cdot M}{d}} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right) \]
   10. Step-by-step derivation

       1. expm1-log1p-u 21.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)\right)} \]

       2. expm1-udef 21.1%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} - 1\right)} \]

       3. *-commutative 21.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} - 1\right) \]

       4. associate-*r/ 24.2%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} - 1\right) \]

   11. Applied egg-rr 24.2%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 24.2%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)\right)} \]

       2. expm1-log1p 24.9%

          \[\leadsto w0 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)} \]

       3. associate-*l* 25.0%

          \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \sqrt{\frac{h}{\ell} \cdot -0.25}\right)\right)} \]

       4. associate-*l/ 25.0%

          \[\leadsto w0 \cdot \left(M \cdot \left(\frac{D}{d} \cdot \sqrt{\color{blue}{\frac{h \cdot -0.25}{\ell}}}\right)\right) \]

   13. Simplified 25.0%

       \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \sqrt{\frac{h \cdot -0.25}{\ell}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \frac{D}{d}\right)\right)\\ \end{array} \]

Alternative 4: 67.6% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 80.2%

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

3. Simplified 80.5%

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

4. Taylor expanded in D around 0 66.6%

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

5. Final simplification 66.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2024024 
(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))))))