Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.9%

Time: 14.6s

Alternatives: 13

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: 80.8% 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: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{+229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) 4e+229)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* 0.5 (/ D d))) 2.0)) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+229) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (0.5 * (D / d))), 2.0)) / l)));
	}
	return tmp;
}

Fortran

NOTE: M 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 ((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 4d+229) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (0.5d0 * (d / d_1))) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+229) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (0.5 * (D / d))), 2.0)) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+229:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (0.5 * (D / d))), 2.0)) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 4e+229)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= 4e+229)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M * (0.5 * (D / d))) ^ 2.0)) / l)));
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[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], 4e+229], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 4e229

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. times-frac 98.3%

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

   3. Simplified 98.3%

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

   if 4e229 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 44.3%

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

      1. *-commutative 44.3%

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

      2. times-frac 48.2%

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

   3. Simplified 48.2%

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

      1. unpow2 48.2%

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

      2. associate-*l* 50.5%

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

      3. div-inv 50.5%

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

      4. associate-*l* 50.4%

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

      5. times-frac 50.4%

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

      6. *-un-lft-identity 50.4%

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

      7. *-commutative 50.4%

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

      8. clear-num 50.4%

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

      9. div-inv 50.4%

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

      10. div-inv 50.4%

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

      11. associate-*l* 50.6%

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

      12. times-frac 50.6%

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

      13. *-un-lft-identity 50.6%

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

      14. *-commutative 50.6%

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

      15. clear-num 50.6%

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

      16. div-inv 50.6%

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

      17. associate-*l* 48.2%

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

   5. Applied egg-rr 72.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{+229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

M = Math.abs(M);
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 ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= Inf)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[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], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

   1. Initial program 92.6%

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

      1. *-commutative 92.6%

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

      2. times-frac 91.3%

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

   3. Simplified 91.3%

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

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in M around 0 74.7%

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

4. Final simplification 89.6%

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

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M 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-19)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))))

C

M = abs(M);
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-19) {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M 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-19) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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-19) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
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-19:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * ((M * D) / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	return tmp

Julia

M = abs(M)
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-19)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
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-19)
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * ((M * D) / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	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: 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-19], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 2e-19

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. times-frac 84.0%

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

   3. Simplified 84.0%

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

      1. unpow2 84.0%

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

      2. associate-*l* 84.4%

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

      3. div-inv 84.4%

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

      4. associate-*l* 84.4%

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

      5. times-frac 84.4%

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

      6. *-un-lft-identity 84.4%

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

      7. *-commutative 84.4%

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

      8. clear-num 84.4%

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

      9. div-inv 84.4%

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

      10. div-inv 84.4%

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

      11. associate-*l* 84.9%

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

      12. times-frac 84.9%

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

      13. *-un-lft-identity 84.9%

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

      14. *-commutative 84.9%

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

      15. clear-num 84.9%

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

      16. div-inv 84.9%

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

      17. associate-*l* 84.5%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in M around 0 91.4%

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

   if 2e-19 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. times-frac 79.6%

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

   3. Simplified 79.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \end{array} \]

Alternative 4: 77.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-132}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.02e-132)
   (* w0 (sqrt (- 1.0 (* (* (/ h (* (/ d M) (/ d M))) (* D (/ D l))) 0.25))))
   (if (<= d 3.2e+70)
     (* w0 (sqrt (- 1.0 (/ (* 0.25 (* h (* (* M D) (* M D)))) (* d (* d l))))))
     w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.02e-132) {
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} else if (d <= 3.2e+70) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (h * ((M * D) * (M * D)))) / (d * (d * 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: 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 (d_1 <= 1.02d-132) then
        tmp = w0 * sqrt((1.0d0 - (((h / ((d_1 / m) * (d_1 / m))) * (d * (d / l))) * 0.25d0)))
    else if (d_1 <= 3.2d+70) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (h * ((m * d) * (m * d)))) / (d_1 * (d_1 * l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 1.02e-132) {
		tmp = w0 * Math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} else if (d <= 3.2e+70) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (h * ((M * D) * (M * D)))) / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.02e-132:
		tmp = w0 * math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)))
	elif d <= 3.2e+70:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (h * ((M * D) * (M * D)))) / (d * (d * l)))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.02e-132)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D * Float64(D / l))) * 0.25))));
	elseif (d <= 3.2e+70)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(h * Float64(Float64(M * D) * Float64(M * D)))) / Float64(d * Float64(d * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.02e-132)
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	elseif (d <= 3.2e+70)
		tmp = w0 * sqrt((1.0 - ((0.25 * (h * ((M * D) * (M * D)))) / (d * (d * 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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.02e-132], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e+70], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(h * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 1.01999999999999998e-132

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 83.0%

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

   3. Simplified 83.0%

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

      1. unpow2 83.0%

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

      2. associate-*l* 83.1%

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

      3. div-inv 83.1%

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

      4. associate-*l* 81.3%

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

      5. times-frac 81.3%

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

      6. *-un-lft-identity 81.3%

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

      7. *-commutative 81.3%

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

      8. clear-num 81.3%

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

      9. div-inv 81.3%

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

      10. div-inv 81.3%

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

      11. associate-*l* 83.2%

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

      12. times-frac 83.2%

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

      13. *-un-lft-identity 83.2%

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

      14. *-commutative 83.2%

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

      15. clear-num 83.2%

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

      16. div-inv 83.2%

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

      17. associate-*l* 83.1%

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

   5. Applied egg-rr 88.9%

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

   6. Taylor expanded in M around 0 48.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative 48.4%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}} \]

      2. times-frac 49.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.25} \]

      3. unpow2 49.7%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      4. associate-*l/ 53.4%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      5. unpow2 53.4%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      6. associate-*r* 59.0%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right) \cdot 0.25} \]

      7. unpow2 59.0%

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

      8. *-commutative 59.0%

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

      9. *-commutative 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in M around 0 53.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]
   10. Step-by-step derivation

       1. *-commutative 53.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       2. associate-/l* 55.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{h}{\frac{{d}^{2}}{{M}^{2}}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       3. unpow2 55.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{\color{blue}{d \cdot d}}{{M}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       4. unpow2 55.3%

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

       5. times-frac 70.8%

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

   11. Simplified 70.8%

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

   if 1.01999999999999998e-132 < d < 3.2000000000000002e70

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. times-frac 78.9%

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

   3. Simplified 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*l* 83.8%

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

      3. div-inv 83.8%

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

      4. associate-*l* 83.8%

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

      5. times-frac 83.8%

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

      6. *-un-lft-identity 83.8%

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

      7. *-commutative 83.8%

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

      8. clear-num 83.7%

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

      9. div-inv 83.8%

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

      10. div-inv 83.8%

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

      11. associate-*l* 83.8%

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

      12. times-frac 83.8%

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

      13. *-un-lft-identity 83.8%

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

      14. *-commutative 83.8%

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

      15. clear-num 83.7%

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

      16. div-inv 83.8%

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

      17. associate-*l* 78.9%

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

   5. Applied egg-rr 82.1%

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

   6. Taylor expanded in M around 0 63.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}} \]

      2. associate-*r* 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{\ell \cdot {d}^{2}}} \]

      3. unpow2 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{\ell \cdot {d}^{2}}} \]

      4. unpow2 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{\ell \cdot {d}^{2}}} \]

      5. unswap-sqr 85.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{\ell \cdot {d}^{2}}} \]

      6. unpow2 85.1%

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

      7. associate-*r* 84.9%

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

   8. Simplified 84.9%

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

   if 3.2000000000000002e70 < d

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. times-frac 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in M around 0 92.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-132}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-135}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{elif}\;d \leq 7.9 \cdot 10^{+68}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d \cdot d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4e-135)
   (* w0 (sqrt (- 1.0 (* (* (/ h (* (/ d M) (/ d M))) (* D (/ D l))) 0.25))))
   (if (<= d 7.9e+68)
     (* w0 (sqrt (- 1.0 (/ (* 0.25 (/ (* h (* (* M D) (* M D))) (* d d))) l))))
     w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4e-135) {
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} else if (d <= 7.9e+68) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((h * ((M * D) * (M * D))) / (d * d))) / 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: 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 (d_1 <= 4d-135) then
        tmp = w0 * sqrt((1.0d0 - (((h / ((d_1 / m) * (d_1 / m))) * (d * (d / l))) * 0.25d0)))
    else if (d_1 <= 7.9d+68) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((h * ((m * d) * (m * d))) / (d_1 * d_1))) / l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 4e-135) {
		tmp = w0 * Math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} else if (d <= 7.9e+68) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((h * ((M * D) * (M * D))) / (d * d))) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4e-135:
		tmp = w0 * math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)))
	elif d <= 7.9e+68:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((h * ((M * D) * (M * D))) / (d * d))) / l)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4e-135)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D * Float64(D / l))) * 0.25))));
	elseif (d <= 7.9e+68)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(h * Float64(Float64(M * D) * Float64(M * D))) / Float64(d * d))) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4e-135)
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	elseif (d <= 7.9e+68)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((h * ((M * D) * (M * D))) / (d * d))) / 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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4e-135], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.9e+68], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(h * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 4.0000000000000002e-135

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 83.0%

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

   3. Simplified 83.0%

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

      1. unpow2 83.0%

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

      2. associate-*l* 83.1%

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

      3. div-inv 83.1%

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

      4. associate-*l* 81.3%

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

      5. times-frac 81.3%

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

      6. *-un-lft-identity 81.3%

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

      7. *-commutative 81.3%

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

      8. clear-num 81.3%

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

      9. div-inv 81.3%

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

      10. div-inv 81.3%

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

      11. associate-*l* 83.2%

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

      12. times-frac 83.2%

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

      13. *-un-lft-identity 83.2%

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

      14. *-commutative 83.2%

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

      15. clear-num 83.2%

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

      16. div-inv 83.2%

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

      17. associate-*l* 83.1%

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

   5. Applied egg-rr 88.9%

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

   6. Taylor expanded in M around 0 48.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative 48.4%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}} \]

      2. times-frac 49.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.25} \]

      3. unpow2 49.7%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      4. associate-*l/ 53.4%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      5. unpow2 53.4%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      6. associate-*r* 59.0%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right) \cdot 0.25} \]

      7. unpow2 59.0%

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

      8. *-commutative 59.0%

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

      9. *-commutative 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in M around 0 53.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]
   10. Step-by-step derivation

       1. *-commutative 53.4%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       2. associate-/l* 55.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{h}{\frac{{d}^{2}}{{M}^{2}}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       3. unpow2 55.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{\color{blue}{d \cdot d}}{{M}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       4. unpow2 55.3%

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

       5. times-frac 70.8%

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

   11. Simplified 70.8%

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

   if 4.0000000000000002e-135 < d < 7.9e68

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. times-frac 78.9%

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

   3. Simplified 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*l* 83.8%

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

      3. div-inv 83.8%

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

      4. associate-*l* 83.8%

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

      5. times-frac 83.8%

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

      6. *-un-lft-identity 83.8%

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

      7. *-commutative 83.8%

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

      8. clear-num 83.7%

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

      9. div-inv 83.8%

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

      10. div-inv 83.8%

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

      11. associate-*l* 83.8%

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

      12. times-frac 83.8%

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

      13. *-un-lft-identity 83.8%

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

      14. *-commutative 83.8%

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

      15. clear-num 83.7%

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

      16. div-inv 83.8%

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

      17. associate-*l* 78.9%

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

   5. Applied egg-rr 82.1%

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

   6. Taylor expanded in M around 0 63.3%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}} \]
   7. Step-by-step derivation

      1. associate-*r* 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}} \]

      2. unpow2 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}}}{\ell}} \]

      3. unpow2 63.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}}}{\ell}} \]

      4. unswap-sqr 85.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}}}{\ell}} \]

      5. unpow2 85.1%

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

   8. Simplified 85.1%

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

   if 7.9e68 < d

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. times-frac 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in M around 0 92.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-135}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{elif}\;d \leq 7.9 \cdot 10^{+68}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d \cdot d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 2.4e+63)
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* D (/ D l)) (* (/ M d) (/ (* M h) d)))))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.4e+63) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (D / l)) * ((M / d) * ((M * h) / d))))));
	} 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: 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 (d_1 <= 2.4d+63) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d * (d / l)) * ((m / d_1) * ((m * h) / d_1))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 2.4e+63) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D * (D / l)) * ((M / d) * ((M * h) / d))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.4e+63:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D * (D / l)) * ((M / d) * ((M * h) / d))))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.4e+63)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M / d) * Float64(Float64(M * h) / d)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.4e+63)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (D / l)) * ((M / d) * ((M * h) / d))))));
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 2.4e+63], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.4e63

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. times-frac 82.8%

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

   3. Simplified 82.8%

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

      1. unpow2 82.8%

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

      2. associate-*l* 83.7%

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

      3. div-inv 83.7%

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

      4. associate-*l* 82.1%

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

      5. times-frac 82.1%

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

      6. *-un-lft-identity 82.1%

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

      7. *-commutative 82.1%

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

      8. clear-num 82.1%

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

      9. div-inv 82.1%

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

      10. div-inv 82.1%

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

      11. associate-*l* 83.8%

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

      12. times-frac 83.7%

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

      13. *-un-lft-identity 83.7%

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

      14. *-commutative 83.7%

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

      15. clear-num 83.7%

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

      16. div-inv 83.7%

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

      17. associate-*l* 82.8%

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

   5. Applied egg-rr 88.2%

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

   6. Taylor expanded in M around 0 51.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative 51.2%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}} \]

      2. times-frac 52.6%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.25} \]

      3. unpow2 52.6%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      4. associate-*l/ 56.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      5. unpow2 56.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      6. associate-*r* 62.1%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right) \cdot 0.25} \]

      7. unpow2 62.1%

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

      8. *-commutative 62.1%

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

      9. *-commutative 62.1%

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

   8. Simplified 62.1%

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

      1. times-frac 72.0%

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

   10. Applied egg-rr 72.0%

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

   if 2.4e63 < d

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 91.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{+63}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.35e+63)
   (* w0 (sqrt (- 1.0 (* (* (/ h (* (/ d M) (/ d M))) (* D (/ D l))) 0.25))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.35e+63) {
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} 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: 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 (d_1 <= 1.35d+63) then
        tmp = w0 * sqrt((1.0d0 - (((h / ((d_1 / m) * (d_1 / m))) * (d * (d / l))) * 0.25d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 1.35e+63) {
		tmp = w0 * Math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.35e+63:
		tmp = w0 * math.sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.35e+63)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D * Float64(D / l))) * 0.25))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.35e+63)
		tmp = w0 * sqrt((1.0 - (((h / ((d / M) * (d / M))) * (D * (D / l))) * 0.25)));
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.35e+63], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.35000000000000009e63

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. times-frac 82.8%

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

   3. Simplified 82.8%

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

      1. unpow2 82.8%

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

      2. associate-*l* 83.7%

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

      3. div-inv 83.7%

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

      4. associate-*l* 82.1%

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

      5. times-frac 82.1%

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

      6. *-un-lft-identity 82.1%

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

      7. *-commutative 82.1%

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

      8. clear-num 82.1%

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

      9. div-inv 82.1%

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

      10. div-inv 82.1%

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

      11. associate-*l* 83.8%

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

      12. times-frac 83.7%

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

      13. *-un-lft-identity 83.7%

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

      14. *-commutative 83.7%

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

      15. clear-num 83.7%

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

      16. div-inv 83.7%

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

      17. associate-*l* 82.8%

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

   5. Applied egg-rr 88.2%

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

   6. Taylor expanded in M around 0 51.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative 51.2%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}} \]

      2. times-frac 52.6%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.25} \]

      3. unpow2 52.6%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      4. associate-*l/ 56.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      5. unpow2 56.3%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.25} \]

      6. associate-*r* 62.1%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{D}{\ell} \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right) \cdot 0.25} \]

      7. unpow2 62.1%

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

      8. *-commutative 62.1%

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

      9. *-commutative 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in M around 0 56.3%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]
   10. Step-by-step derivation

       1. *-commutative 56.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       2. associate-/l* 57.9%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{h}{\frac{{d}^{2}}{{M}^{2}}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       3. unpow2 57.9%

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{\color{blue}{d \cdot d}}{{M}^{2}}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25} \]

       4. unpow2 57.9%

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

       5. times-frac 70.9%

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

   11. Simplified 70.9%

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

   if 1.35000000000000009e63 < d

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 91.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{+63}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \left(D \cdot \frac{D}{\ell}\right)\right) \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot \left(\frac{M \cdot w0}{\frac{d}{M}} \cdot \frac{h}{d}\right)}{\frac{\ell}{D}}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 5.2e+96)
   (fma -0.125 (/ (* D (* (/ (* M w0) (/ d M)) (/ h d))) (/ l D)) w0)
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.2e+96) {
		tmp = fma(-0.125, ((D * (((M * w0) / (d / M)) * (h / d))) / (l / D)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.2e+96)
		tmp = fma(-0.125, Float64(Float64(D * Float64(Float64(Float64(M * w0) / Float64(d / M)) * Float64(h / d))) / Float64(l / D)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: M 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[d, 5.2e+96], N[(-0.125 * N[(N[(D * N[(N[(N[(M * w0), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 5.2e96

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around 0 48.8%

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

      1. fma-def 48.8%

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

      2. times-frac 50.2%

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

      3. unpow2 50.2%

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

      4. associate-/l* 53.7%

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

      5. *-commutative 53.7%

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

      6. associate-*r* 54.2%

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

      7. unpow2 54.2%

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

      8. times-frac 58.9%

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

      9. unpow2 58.9%

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

      10. associate-*r* 60.9%

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

      11. *-commutative 60.9%

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

   6. Simplified 60.9%

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

      1. associate-*l/ 64.6%

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

      2. associate-/l* 68.6%

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

   8. Applied egg-rr 68.6%

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

   if 5.2e96 < d

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. times-frac 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 91.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot \left(\frac{M \cdot w0}{\frac{d}{M}} \cdot \frac{h}{d}\right)}{\frac{\ell}{D}}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 73.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.15e-11)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (/ (* M (* M h)) (* d d))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.15e-11) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (M * h)) / (d * d)))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 1.15d-11) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((m * (m * h)) / (d_1 * d_1)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 <= 1.15e-11) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (M * h)) / (d * d)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.15e-11:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (M * h)) / (d * d)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.15e-11)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M * Float64(M * h)) / Float64(d * d))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.15e-11)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M * (M * h)) / (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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.15e-11], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.15000000000000007e-11

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. times-frac 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in M around 0 77.8%

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

   if 1.15000000000000007e-11 < M

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 39.5%

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

      1. *-commutative 39.5%

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

      2. unpow2 39.5%

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

      3. unpow2 39.5%

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

      4. unpow2 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in D around 0 39.5%

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

      1. times-frac 36.1%

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

      2. unpow2 36.1%

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

      3. associate-*l/ 37.9%

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

      4. unpow2 37.9%

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

      5. associate-*r* 54.8%

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

      6. unpow2 54.8%

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

      7. *-commutative 54.8%

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

      8. *-commutative 54.8%

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

   9. Simplified 54.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right)\right)\\ \end{array} \]

Alternative 10: 71.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{+68}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.35e+68)
   (* w0 (+ 1.0 (* -0.125 (/ (* h (* (* M D) (* M D))) (* d (* d l))))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.35e+68) {
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M * D) * (M * D))) / (d * (d * 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: 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 (d_1 <= 1.35d+68) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h * ((m * d) * (m * d))) / (d_1 * (d_1 * l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 1.35e+68) {
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M * D) * (M * D))) / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.35e+68:
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M * D) * (M * D))) / (d * (d * l)))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.35e+68)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * Float64(Float64(M * D) * Float64(M * D))) / Float64(d * Float64(d * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.35e+68)
		tmp = w0 * (1.0 + (-0.125 * ((h * ((M * D) * (M * D))) / (d * (d * 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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.35e+68], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.34999999999999995e68

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in M around 0 49.5%

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

      1. *-commutative 49.5%

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

      2. unpow2 49.5%

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

      3. unpow2 49.5%

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

      4. unpow2 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in D around 0 49.5%

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

      1. associate-*r* 52.1%

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

      2. unpow2 52.1%

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

      3. unpow2 52.1%

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

      4. unswap-sqr 65.3%

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

      5. unpow2 65.3%

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

      6. associate-*r* 70.2%

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

   9. Simplified 70.2%

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

   if 1.34999999999999995e68 < d

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. times-frac 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in M around 0 92.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{+68}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 72.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.25 \cdot 10^{+63}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 2.25e+63)
   (+ w0 (* w0 (* (* (/ M d) (/ (* M h) d)) (* (* D (/ D l)) -0.125))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.25e+63) {
		tmp = w0 + (w0 * (((M / d) * ((M * h) / d)) * ((D * (D / l)) * -0.125)));
	} 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: 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 (d_1 <= 2.25d+63) then
        tmp = w0 + (w0 * (((m / d_1) * ((m * h) / d_1)) * ((d * (d / l)) * (-0.125d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 (d <= 2.25e+63) {
		tmp = w0 + (w0 * (((M / d) * ((M * h) / d)) * ((D * (D / l)) * -0.125)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.25e+63:
		tmp = w0 + (w0 * (((M / d) * ((M * h) / d)) * ((D * (D / l)) * -0.125)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.25e+63)
		tmp = Float64(w0 + Float64(w0 * Float64(Float64(Float64(M / d) * Float64(Float64(M * h) / d)) * Float64(Float64(D * Float64(D / l)) * -0.125))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.25e+63)
		tmp = w0 + (w0 * (((M / d) * ((M * h) / d)) * ((D * (D / l)) * -0.125)));
	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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 2.25e+63], N[(w0 + N[(w0 * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.25000000000000008e63

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. times-frac 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in M around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. unpow2 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in D around 0 49.8%

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

      1. times-frac 50.7%

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

      2. unpow2 50.7%

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

      3. associate-*l/ 54.4%

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

      4. unpow2 54.4%

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

      5. associate-*r* 60.1%

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

      6. unpow2 60.1%

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

      7. *-commutative 60.1%

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

      8. *-commutative 60.1%

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

   9. Simplified 60.1%

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

       1. distribute-rgt-in 60.1%

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

       2. *-un-lft-identity 60.1%

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

       3. associate-*l* 60.1%

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

       4. times-frac 68.8%

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

   11. Applied egg-rr 68.8%

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

   if 2.25000000000000008e63 < d

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 91.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.25 \cdot 10^{+63}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 12: 71.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h \cdot w0}{d} \cdot \frac{M \cdot M}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e+31)
   w0
   (* -0.125 (* (* D (/ D l)) (* (/ (* h w0) d) (/ (* M M) d))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((h * w0) / d) * ((M * M) / d)));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 9d+31) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * (((h * w0) / d_1) * ((m * m) / d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 <= 9e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((h * w0) / d) * ((M * M) / d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e+31:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * (((h * w0) / d) * ((M * M) / d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e+31)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(h * w0) / d) * Float64(Float64(M * M) / d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e+31)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * (((h * w0) / d) * ((M * M) / 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: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e+31], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * w0), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999992e31

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. times-frac 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in M around 0 76.9%

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

   if 8.9999999999999992e31 < M

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. times-frac 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in M around 0 36.6%

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

      1. *-commutative 36.6%

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

      2. unpow2 36.6%

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

      3. unpow2 36.6%

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

      4. unpow2 36.6%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in D around 0 36.6%

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

      1. times-frac 32.5%

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

      2. unpow2 32.5%

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

      3. associate-*l/ 34.6%

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

      4. unpow2 34.6%

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

      5. associate-*r* 54.5%

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

      6. unpow2 54.5%

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

      7. *-commutative 54.5%

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

      8. *-commutative 54.5%

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

   9. Simplified 54.5%

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

   10. Taylor expanded in M around inf 23.3%

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

       1. *-commutative 23.3%

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

       2. *-commutative 23.3%

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

       3. *-commutative 23.3%

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

       4. times-frac 21.1%

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

       5. unpow2 21.1%

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

       6. associate-*r/ 23.2%

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

       7. *-commutative 23.2%

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

       8. associate-*r* 23.2%

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

       9. unpow2 23.2%

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

       10. unpow2 23.2%

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

   12. Simplified 23.2%

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

       1. times-frac 25.6%

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

       2. *-commutative 25.6%

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

   14. Applied egg-rr 25.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h \cdot w0}{d} \cdot \frac{M \cdot M}{d}\right)\right)\\ \end{array} \]

Alternative 13: 67.5% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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);
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: 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);
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)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M = abs(M)
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)
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: 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 83.2%

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

   1. *-commutative 83.2%

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

   2. times-frac 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in M around 0 72.9%

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

5. Final simplification 72.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023215 
(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))))))