Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.3% → 87.5%

Time: 15.7s

Alternatives: 13

Speedup: 1.9×

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: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 5e+112)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* (* M 0.5) (/ D d)) 2.0) h) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112) {
		tmp = w0 * sqrt((1.0 - ((pow(((M * 0.5) * (D / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 5d+112) then
        tmp = w0 * sqrt((1.0d0 - (((((m * 0.5d0) * (d / d_1)) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow(((M * 0.5) * (D / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112:
		tmp = w0 * math.sqrt((1.0 - ((math.pow(((M * 0.5) * (D / d)), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	return tmp

Julia

M = abs(M)
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)) ^ 2.0) <= 5e+112)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 5e+112)
		tmp = w0 * sqrt((1.0 - (((((M * 0.5) * (D / d)) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+112], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5e112

   1. Initial program 93.4%

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

      1. times-frac 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.2%

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

      3. metadata-eval 96.2%

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

   5. Applied egg-rr 96.2%

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

   if 5e112 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 57.3%

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

      1. times-frac 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in w0 around 0 43.6%

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

      1. cancel-sign-sub-inv 43.6%

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

      2. *-commutative 43.6%

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

      3. cancel-sign-sub-inv 43.6%

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

   6. Simplified 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*r* 44.1%

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

      3. times-frac 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. *-lft-identity 54.8%

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

      2. associate-*l* 54.8%

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

      3. associate-*r* 57.6%

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

      4. associate-/l* 61.0%

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

   10. Simplified 61.0%

       \[\leadsto \color{blue}{\sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right) \cdot \frac{M}{d}\right)}} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]

Alternative 2: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+261}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M 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))) 1e+261)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))))

C

M = abs(M);
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))) <= 1e+261) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Fortran

NOTE: M 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))) <= 1d+261) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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))) <= 1e+261) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Python

M = abs(M)
[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))) <= 1e+261:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	return tmp

Julia

M = abs(M)
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))) <= 1e+261)
		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(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
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))) <= 1e+261)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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], 1e+261], 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[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999993e260

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. times-frac 98.9%

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

   3. Simplified 98.9%

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

   if 9.9999999999999993e260 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 46.5%

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

      1. times-frac 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in w0 around 0 42.7%

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

      1. cancel-sign-sub-inv 42.7%

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

      2. *-commutative 42.7%

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

      3. cancel-sign-sub-inv 42.7%

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

   6. Simplified 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. associate-*r* 41.4%

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

      3. times-frac 52.7%

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

   8. Applied egg-rr 52.7%

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

      1. *-lft-identity 52.7%

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

      2. associate-*l* 52.7%

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

      3. associate-*r* 53.1%

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

      4. associate-/l* 56.0%

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

   10. Simplified 56.0%

       \[\leadsto \color{blue}{\sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right) \cdot \frac{M}{d}\right)}} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+261}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+261}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M 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))) 1e+261)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (/ M d) (/ 2.0 D)) 2.0)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))))

C

M = abs(M);
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))) <= 1e+261) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Fortran

NOTE: M 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))) <= 1d+261) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) / (2.0d0 / d)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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))) <= 1e+261) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Python

M = abs(M)
[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))) <= 1e+261:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) / (2.0 / D)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	return tmp

Julia

M = abs(M)
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))) <= 1e+261)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) / Float64(2.0 / D)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
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))) <= 1e+261)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) / (2.0 / D)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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], 1e+261], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] / N[(2.0 / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999993e260

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. times-frac 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 98.9%

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

      2. un-div-inv 98.9%

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

   5. Applied egg-rr 98.9%

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

   if 9.9999999999999993e260 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 46.5%

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

      1. times-frac 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in w0 around 0 42.7%

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

      1. cancel-sign-sub-inv 42.7%

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

      2. *-commutative 42.7%

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

      3. cancel-sign-sub-inv 42.7%

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

   6. Simplified 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. associate-*r* 41.4%

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

      3. times-frac 52.7%

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

   8. Applied egg-rr 52.7%

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

      1. *-lft-identity 52.7%

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

      2. associate-*l* 52.7%

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

      3. associate-*r* 53.1%

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

      4. associate-/l* 56.0%

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

   10. Simplified 56.0%

       \[\leadsto \color{blue}{\sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right) \cdot \frac{M}{d}\right)}} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+261}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]

Alternative 4: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 5e+112)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ D (/ d M))) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112) {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * (D / (d / M))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 5d+112) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * (d / (d_1 / m))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * (D / (d / M))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+112:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * (D / (d / M))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	return tmp

Julia

M = abs(M)
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)) ^ 2.0) <= 5e+112)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(D / Float64(d / M))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 5e+112)
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * (D / (d / M))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+112], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5e112

   1. Initial program 93.4%

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

      1. times-frac 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.2%

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

      3. metadata-eval 96.2%

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

   5. Applied egg-rr 96.2%

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

   6. Taylor expanded in M around 0 96.7%

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

      1. associate-/l* 96.2%

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

   8. Simplified 96.2%

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

   if 5e112 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 57.3%

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

      1. times-frac 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in w0 around 0 43.6%

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

      1. cancel-sign-sub-inv 43.6%

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

      2. *-commutative 43.6%

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

      3. cancel-sign-sub-inv 43.6%

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

   6. Simplified 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*r* 44.1%

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

      3. times-frac 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. *-lft-identity 54.8%

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

      2. associate-*l* 54.8%

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

      3. associate-*r* 57.6%

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

      4. associate-/l* 61.0%

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

   10. Simplified 61.0%

       \[\leadsto \color{blue}{\sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right) \cdot \frac{M}{d}\right)}} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]

Alternative 5: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-65}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 2e-65)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2e-65) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 2d-65) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / d_1) * (m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2e-65) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2e-65:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2e-65)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2e-65)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 2e-65], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.99999999999999985e-65

   1. Initial program 78.3%

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

      1. times-frac 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in w0 around 0 49.5%

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

      1. cancel-sign-sub-inv 49.5%

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

      2. *-commutative 49.5%

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

      3. cancel-sign-sub-inv 49.5%

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

   6. Simplified 50.0%

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

      1. *-un-lft-identity 50.0%

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

      2. associate-*r* 50.0%

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

      3. times-frac 65.6%

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

   8. Applied egg-rr 65.6%

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

      1. *-lft-identity 65.6%

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

      2. associate-*l* 65.6%

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

      3. associate-*r* 70.5%

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

      4. associate-/l* 73.1%

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

   10. Simplified 73.1%

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

   if 1.99999999999999985e-65 < d

   1. Initial program 85.9%

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

      1. times-frac 87.1%

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

   3. Simplified 87.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-65}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \end{array} \]

Alternative 6: 79.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{-64}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 5.3e-64)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ M d) (* (/ D (/ l (* D h))) (/ M d)))))))
   (* w0 (+ 1.0 (* -0.125 (* h (/ (pow (/ D (/ d M)) 2.0) l)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.3e-64) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (pow((D / (d / M)), 2.0) / l))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 5.3d-64) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((m / d_1) * ((d / (l / (d * h))) * (m / d_1))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((d / (d_1 / m)) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.3e-64) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (Math.pow((D / (d / M)), 2.0) / l))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.3e-64:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * (math.pow((D / (d / M)), 2.0) / l))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.3e-64)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(D / Float64(l / Float64(D * h))) * Float64(M / d)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64((Float64(D / Float64(d / M)) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.3e-64)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((M / d) * ((D / (l / (D * h))) * (M / d))))));
	else
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / (d / M)) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 5.3e-64], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.3000000000000002e-64

   1. Initial program 78.3%

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

      1. times-frac 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in w0 around 0 49.5%

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

      1. cancel-sign-sub-inv 49.5%

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

      2. *-commutative 49.5%

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

      3. cancel-sign-sub-inv 49.5%

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

   6. Simplified 50.0%

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

      1. *-un-lft-identity 50.0%

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

      2. associate-*r* 50.0%

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

      3. times-frac 65.6%

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

   8. Applied egg-rr 65.6%

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

      1. *-lft-identity 65.6%

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

      2. associate-*l* 65.6%

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

      3. associate-*r* 70.5%

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

      4. associate-/l* 73.1%

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

   10. Simplified 73.1%

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

   if 5.3000000000000002e-64 < d

   1. Initial program 85.9%

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

      1. times-frac 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in M around 0 55.3%

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

      1. associate-*r/ 55.3%

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

      2. *-commutative 55.3%

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

      3. associate-*r/ 55.3%

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

      4. associate-/l* 55.3%

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

      5. unpow2 55.3%

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

      6. unpow2 55.3%

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

      7. *-commutative 55.3%

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

      8. unpow2 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in l around 0 55.3%

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

      1. unpow2 55.3%

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

      2. unpow2 55.3%

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

      3. *-commutative 55.3%

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

      4. times-frac 55.3%

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

      5. times-frac 62.9%

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

   9. Simplified 62.9%

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

       1. times-frac 73.7%

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

       2. pow2 73.7%

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

   11. Applied egg-rr 73.7%

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

       1. frac-times 62.9%

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

   13. Applied egg-rr 62.9%

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

       1. times-frac 73.7%

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

       2. associate-*l/ 77.6%

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

       3. associate-/r/ 78.8%

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

       4. associate-*r/ 64.3%

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

       5. unpow2 64.3%

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

       6. associate-/r* 68.3%

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

       7. associate-*r/ 81.4%

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

       8. associate-/l* 81.4%

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

       9. *-commutative 81.4%

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

       10. associate-*l/ 81.4%

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

       11. associate-*l/ 82.6%

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

       12. associate-/l* 81.4%

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

       13. *-commutative 81.4%

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

       14. associate-*l/ 82.6%

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

       15. unpow2 82.6%

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

       16. *-commutative 82.6%

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

   15. Simplified 82.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{-64}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{D}{\frac{\ell}{D \cdot h}} \cdot \frac{M}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]

Alternative 7: 80.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right) \end{array} \]

FPCore

NOTE: M 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 (+ 1.0 (* -0.125 (* h (/ (pow (/ D (/ d M)) 2.0) l))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (h * (pow((D / (d / M)), 2.0) / l))));
}

Fortran

NOTE: M 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 * (1.0d0 + ((-0.125d0) * (h * (((d / (d_1 / m)) ** 2.0d0) / l))))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (h * (Math.pow((D / (d / M)), 2.0) / l))));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * (h * (math.pow((D / (d / M)), 2.0) / l))))

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64((Float64(D / Float64(d / M)) ^ 2.0) / l)))))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * (h * (((D / (d / M)) ^ 2.0) / l))));
end

Wolfram

NOTE: M 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_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

   1. times-frac 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in M around 0 50.6%

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

   1. associate-*r/ 50.6%

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

   2. *-commutative 50.6%

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

   3. associate-*r/ 50.6%

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

   4. associate-/l* 50.7%

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

   5. unpow2 50.7%

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

   6. unpow2 50.7%

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

   7. *-commutative 50.7%

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

   8. unpow2 50.7%

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

6. Simplified 50.7%

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

7. Taylor expanded in l around 0 50.7%

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

   1. unpow2 50.7%

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

   2. unpow2 50.7%

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

   3. *-commutative 50.7%

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

   4. times-frac 49.9%

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

   5. times-frac 60.7%

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

9. Simplified 60.7%

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

    1. times-frac 68.8%

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

    2. pow2 68.8%

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

11. Applied egg-rr 68.8%

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

    1. frac-times 60.7%

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

13. Applied egg-rr 60.7%

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

    1. times-frac 68.8%

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

    2. associate-*l/ 71.7%

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

    3. associate-/r/ 74.3%

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

    4. associate-*r/ 63.9%

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

    5. unpow2 63.9%

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

    6. associate-/r* 66.9%

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

    7. associate-*r/ 75.6%

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

    8. associate-/l* 75.6%

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

    9. *-commutative 75.6%

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

    10. associate-*l/ 75.6%

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

    11. associate-*l/ 77.2%

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

    12. associate-/l* 76.4%

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

    13. *-commutative 76.4%

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

    14. associate-*l/ 77.2%

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

    15. unpow2 77.2%

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

    16. *-commutative 77.2%

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

15. Simplified 77.2%

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

16. Final simplification 77.2%

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

Alternative 8: 68.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := M \cdot \left(M \cdot h\right)\\ \mathbf{if}\;d \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{\ell}{\frac{t_0}{d \cdot d}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(t_0 \cdot \frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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
 (let* ((t_0 (* M (* M h))))
   (if (<= d 2.25e-178)
     (* w0 (+ 1.0 (* -0.125 (/ (* D D) (* (/ l h) (* (/ d M) (/ d M)))))))
     (if (<= d 2.05e+124)
       (* w0 (+ 1.0 (* -0.125 (/ D (/ (/ l (/ t_0 (* d d))) D)))))
       (* w0 (+ 1.0 (* -0.125 (* t_0 (/ (* (/ D d) (/ D d)) l)))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (M * h);
	double tmp;
	if (d <= 2.25e-178) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	} else if (d <= 2.05e+124) {
		tmp = w0 * (1.0 + (-0.125 * (D / ((l / (t_0 / (d * d))) / D))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (((D / d) * (D / d)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M 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) :: t_0
    real(8) :: tmp
    t_0 = m * (m * h)
    if (d_1 <= 2.25d-178) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) / ((l / h) * ((d_1 / m) * (d_1 / m))))))
    else if (d_1 <= 2.05d+124) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d / ((l / (t_0 / (d_1 * d_1))) / d))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (t_0 * (((d / d_1) * (d / d_1)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (M * h);
	double tmp;
	if (d <= 2.25e-178) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	} else if (d <= 2.05e+124) {
		tmp = w0 * (1.0 + (-0.125 * (D / ((l / (t_0 / (d * d))) / D))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (((D / d) * (D / d)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = M * (M * h)
	tmp = 0
	if d <= 2.25e-178:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))))
	elif d <= 2.05e+124:
		tmp = w0 * (1.0 + (-0.125 * (D / ((l / (t_0 / (d * d))) / D))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (((D / d) * (D / d)) / l))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(M * h))
	tmp = 0.0
	if (d <= 2.25e-178)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(l / h) * Float64(Float64(d / M) * Float64(d / M)))))));
	elseif (d <= 2.05e+124)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D / Float64(Float64(l / Float64(t_0 / Float64(d * d))) / D)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(t_0 * Float64(Float64(Float64(D / d) * Float64(D / d)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M * (M * h);
	tmp = 0.0;
	if (d <= 2.25e-178)
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	elseif (d <= 2.05e+124)
		tmp = w0 * (1.0 + (-0.125 * (D / ((l / (t_0 / (d * d))) / D))));
	else
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (((D / d) * (D / d)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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_] := Block[{t$95$0 = N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2.25e-178], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e+124], N[(w0 * N[(1.0 + N[(-0.125 * N[(D / N[(N[(l / N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(t$95$0 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.24999999999999989e-178

   1. Initial program 78.5%

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

      1. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 46.3%

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

      1. associate-*r/ 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-*r/ 46.3%

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

      4. associate-/l* 46.4%

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

      5. unpow2 46.4%

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

      6. unpow2 46.4%

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

      7. *-commutative 46.4%

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

      8. unpow2 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in l around 0 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. *-commutative 46.4%

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

      4. times-frac 45.7%

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

      5. times-frac 60.0%

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

   9. Simplified 60.0%

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

   if 2.24999999999999989e-178 < d < 2.05000000000000001e124

   1. Initial program 81.0%

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

      1. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in M around 0 55.8%

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

      1. associate-*r/ 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*r/ 55.8%

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

      4. associate-/l* 55.8%

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

      5. unpow2 55.8%

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

      6. unpow2 55.8%

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

      7. *-commutative 55.8%

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

      8. unpow2 55.8%

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

   6. Simplified 55.8%

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

      1. *-un-lft-identity 55.8%

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

      2. associate-/l* 68.7%

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

      3. associate-/l* 69.6%

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

      4. associate-*l* 72.7%

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

   8. Applied egg-rr 72.7%

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

   if 2.05000000000000001e124 < d

   1. Initial program 89.1%

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

      1. times-frac 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in M around 0 58.3%

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

      1. associate-*r/ 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*r/ 58.3%

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

      4. associate-/l* 58.3%

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

      5. unpow2 58.3%

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

      6. unpow2 58.3%

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

      7. *-commutative 58.3%

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

      8. unpow2 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in D around 0 58.3%

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

      1. unpow2 58.3%

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

      2. *-commutative 58.3%

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

      3. *-commutative 58.3%

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

      4. unpow2 58.3%

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

      5. associate-*r* 61.1%

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

      6. associate-*r/ 61.1%

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

      7. *-commutative 61.1%

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

      8. associate-/r* 61.1%

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

      9. unpow2 61.1%

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

      10. times-frac 78.6%

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

      11. unpow2 78.6%

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

   9. Simplified 78.6%

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

       1. unpow2 78.6%

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

   11. Applied egg-rr 78.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{\ell}{\frac{M \cdot \left(M \cdot h\right)}{d \cdot d}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell}\right)\right)\\ \end{array} \]

Alternative 9: 72.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.6 \cdot 10^{-121}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 3.6e-121)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* M (* M h)) (/ (* (/ D d) (/ D d)) l)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.6e-121) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((M * (M * h)) * (((D / d) * (D / d)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 3.6d-121) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((m * (m * h)) * (((d / d_1) * (d / d_1)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.6e-121) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((M * (M * h)) * (((D / d) * (D / d)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.6e-121:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((M * (M * h)) * (((D / d) * (D / d)) / l))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.6e-121)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(M * Float64(M * h)) * Float64(Float64(Float64(D / d) * Float64(D / d)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.6e-121)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((M * (M * h)) * (((D / d) * (D / d)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 3.6e-121], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.59999999999999984e-121

   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. times-frac 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in M around 0 67.1%

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

   if 3.59999999999999984e-121 < M

   1. Initial program 74.1%

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

      1. times-frac 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in M around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-*r/ 44.8%

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

      4. associate-/l* 45.0%

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

      5. unpow2 45.0%

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

      6. unpow2 45.0%

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

      7. *-commutative 45.0%

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

      8. unpow2 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in D around 0 44.8%

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

      1. unpow2 44.8%

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

      2. *-commutative 44.8%

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

      3. *-commutative 44.8%

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

      4. unpow2 44.8%

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

      5. associate-*r* 48.4%

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

      6. associate-*r/ 48.6%

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

      7. *-commutative 48.6%

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

      8. associate-/r* 48.6%

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

      9. unpow2 48.6%

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

      10. times-frac 57.3%

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

      11. unpow2 57.3%

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

   9. Simplified 57.3%

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

       1. unpow2 57.3%

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

   11. Applied egg-rr 57.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.6 \cdot 10^{-121}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\frac{D}{d} \cdot \frac{D}{d}}{\ell}\right)\right)\\ \end{array} \]

Alternative 10: 69.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 1.95e-13)
   w0
   (* -0.125 (* (* D (/ D l)) (/ (* w0 (* h (* M M))) (* d d))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.95e-13) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 1.95d-13) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * ((w0 * (h * (m * m))) / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.95e-13) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.95e-13:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.95e-13)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / Float64(d * d))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.95e-13)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 1.95e-13], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.95000000000000002e-13

   1. Initial program 84.0%

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

      1. times-frac 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 67.5%

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

   if 1.95000000000000002e-13 < M

   1. Initial program 71.9%

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

      1. times-frac 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*r/ 38.3%

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

      4. associate-/l* 38.6%

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

      5. unpow2 38.6%

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

      6. unpow2 38.6%

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

      7. *-commutative 38.6%

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

      8. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in l around 0 38.6%

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

      1. unpow2 38.6%

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

      2. unpow2 38.6%

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

      3. *-commutative 38.6%

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

      4. times-frac 37.2%

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

      5. times-frac 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in D around inf 20.1%

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

       1. times-frac 20.1%

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

       2. unpow2 20.1%

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

       3. unpow2 20.1%

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

       4. *-commutative 20.1%

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

       5. unpow2 20.1%

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

   12. Simplified 20.1%

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

   13. Taylor expanded in D around 0 20.1%

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

       1. unpow2 20.1%

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

       2. associate-*l/ 21.7%

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

       3. *-commutative 21.7%

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

   15. Simplified 21.7%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 11: 69.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{w0}{d} \cdot \frac{M \cdot h}{\frac{d}{M}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 2.6e-15)
   w0
   (* -0.125 (* (/ (* D D) l) (* (/ w0 d) (/ (* M h) (/ d M)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.6e-15) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((M * h) / (d / M))));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 2.6d-15) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / l) * ((w0 / d_1) * ((m * h) / (d_1 / m))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.6e-15) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((M * h) / (d / M))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.6e-15:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((M * h) / (d / M))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.6e-15)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(w0 / d) * Float64(Float64(M * h) / Float64(d / M)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.6e-15)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((M * h) / (d / M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 2.6e-15], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(w0 / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.60000000000000004e-15

   1. Initial program 84.0%

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

      1. times-frac 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 67.5%

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

   if 2.60000000000000004e-15 < M

   1. Initial program 71.9%

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

      1. times-frac 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*r/ 38.3%

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

      4. associate-/l* 38.6%

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

      5. unpow2 38.6%

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

      6. unpow2 38.6%

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

      7. *-commutative 38.6%

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

      8. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in l around 0 38.6%

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

      1. unpow2 38.6%

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

      2. unpow2 38.6%

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

      3. *-commutative 38.6%

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

      4. times-frac 37.2%

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

      5. times-frac 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in D around inf 20.1%

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

       1. times-frac 20.1%

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

       2. unpow2 20.1%

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

       3. unpow2 20.1%

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

       4. *-commutative 20.1%

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

       5. unpow2 20.1%

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

   12. Simplified 20.1%

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

   13. Taylor expanded in h around 0 20.1%

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

       1. *-commutative 20.1%

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

       2. unpow2 20.1%

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

       3. associate-*l* 20.2%

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

       4. unpow2 20.2%

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

       5. associate-/r* 22.7%

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

       6. *-commutative 22.7%

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

       7. associate-*r/ 24.3%

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

       8. associate-*l/ 24.3%

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

       9. associate-/l* 24.3%

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

       10. *-commutative 24.3%

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

   15. Simplified 24.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{w0}{d} \cdot \frac{M \cdot h}{\frac{d}{M}}\right)\right)\\ \end{array} \]

Alternative 12: 69.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{w0}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M 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 9.2e-12)
   w0
   (* -0.125 (* (/ (/ D (/ l D)) d) (* (* M (* M h)) (/ w0 d))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9.2e-12) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / (l / D)) / d) * ((M * (M * h)) * (w0 / d)));
	}
	return tmp;
}

Fortran

NOTE: M 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 <= 9.2d-12) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / (l / d)) / d_1) * ((m * (m * h)) * (w0 / d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9.2e-12) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / (l / D)) / d) * ((M * (M * h)) * (w0 / d)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9.2e-12:
		tmp = w0
	else:
		tmp = -0.125 * (((D / (l / D)) / d) * ((M * (M * h)) * (w0 / d)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9.2e-12)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / Float64(l / D)) / d) * Float64(Float64(M * Float64(M * h)) * Float64(w0 / d))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9.2e-12)
		tmp = w0;
	else
		tmp = -0.125 * (((D / (l / D)) / d) * ((M * (M * h)) * (w0 / d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M 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, 9.2e-12], w0, N[(-0.125 * N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.19999999999999957e-12

   1. Initial program 84.0%

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

      1. times-frac 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 67.5%

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

   if 9.19999999999999957e-12 < M

   1. Initial program 71.9%

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

      1. times-frac 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*r/ 38.3%

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

      4. associate-/l* 38.6%

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

      5. unpow2 38.6%

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

      6. unpow2 38.6%

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

      7. *-commutative 38.6%

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

      8. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in l around 0 38.6%

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

      1. unpow2 38.6%

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

      2. unpow2 38.6%

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

      3. *-commutative 38.6%

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

      4. times-frac 37.2%

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

      5. times-frac 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in D around inf 20.1%

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

       1. times-frac 20.1%

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

       2. unpow2 20.1%

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

       3. unpow2 20.1%

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

       4. *-commutative 20.1%

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

       5. unpow2 20.1%

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

   12. Simplified 20.1%

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

   13. Taylor expanded in D around 0 20.1%

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

       1. *-commutative 20.1%

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

       2. unpow2 20.1%

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

       3. times-frac 20.1%

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

       4. *-commutative 20.1%

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

       5. *-commutative 20.1%

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

       6. unpow2 20.1%

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

       7. associate-*l* 20.2%

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

       8. associate-*r/ 20.2%

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

       9. unpow2 20.2%

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

       10. times-frac 22.7%

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

       11. *-commutative 22.7%

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

       12. associate-*r/ 24.3%

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

   15. Simplified 25.8%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{\frac{D}{\frac{\ell}{D}}}{d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{w0}{d}\right)\right)\\ \end{array} \]

Alternative 13: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 80.7%

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

   1. times-frac 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in M around 0 60.6%

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

5. Final simplification 60.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023240 
(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))))))