Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.6%

Time: 14.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l))))
   (if (<= t_0 5e-8) (* w0 (sqrt (- 1.0 t_0))) w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (d * 2.0)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)
    if (t_0 <= 5d-8) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 5e-8:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (d * 2.0)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 5e-8)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 4.9999999999999998e-8

   1. Initial program 90.9%

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

   if 4.9999999999999998e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 62.9%

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

4. Final simplification 89.0%

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

Alternative 2: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -5e-295)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-295) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((h / l) <= (-5d-295)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / 2.0d0) * (d / d_1)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-295) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e-295:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-295)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-295)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-295], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.00000000000000008e-295

   1. Initial program 82.4%

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

      1. times-frac 82.4%

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

   3. Simplified 82.4%

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

   if -5.00000000000000008e-295 < (/.f64 h l)

   1. Initial program 87.8%

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

      1. times-frac 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in M around 0 94.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* (pow (* M (* D (/ 0.5 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 * (0.5 / 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 * (0.5d0 / 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 * (0.5 / d))), 2.0) * h) / l)));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((math.pow((M * (D * (0.5 / 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 * Float64(D * Float64(0.5 / d))) ^ 2.0) * h) / l))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * (D * (0.5 / d))) ^ 2.0) * h) / l)));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

   1. times-frac 84.8%

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

3. Simplified 84.8%

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

   1. unpow2 84.8%

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

   2. unpow2 84.8%

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

   3. frac-times 84.8%

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

   4. associate-*r/ 87.3%

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

   5. frac-times 87.3%

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

   6. unpow2 87.3%

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

   7. unpow2 87.3%

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

   8. frac-times 87.3%

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

   9. div-inv 87.3%

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

   10. associate-*l* 87.3%

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

   11. associate-/r* 87.3%

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

   12. metadata-eval 87.3%

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

5. Applied egg-rr 87.3%

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

6. Final simplification 87.3%

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

Alternative 4: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot {\left(\frac{d}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0
          (sqrt
           (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))))
   (if (<= (/ h l) -1e+138)
     t_0
     (if (<= (/ h l) -5e-8)
       (* w0 (+ 1.0 (* (/ (* D h) (* (/ l D) (pow (/ d M) 2.0))) -0.125)))
       (if (<= (/ h l) -2e-182) t_0 w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = t_0;
	} else if ((h / l) <= -5e-8) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * pow((d / M), 2.0))) * -0.125));
	} else if ((h / l) <= -2e-182) {
		tmp = t_0;
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)))))
    if ((h / l) <= (-1d+138)) then
        tmp = t_0
    else if ((h / l) <= (-5d-8)) then
        tmp = w0 * (1.0d0 + (((d * h) / ((l / d) * ((d_1 / m) ** 2.0d0))) * (-0.125d0)))
    else if ((h / l) <= (-2d-182)) then
        tmp = t_0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = t_0;
	} else if ((h / l) <= -5e-8) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * Math.pow((d / M), 2.0))) * -0.125));
	} else if ((h / l) <= -2e-182) {
		tmp = t_0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	tmp = 0
	if (h / l) <= -1e+138:
		tmp = t_0
	elif (h / l) <= -5e-8:
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * math.pow((d / M), 2.0))) * -0.125))
	elif (h / l) <= -2e-182:
		tmp = t_0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))))
	tmp = 0.0
	if (Float64(h / l) <= -1e+138)
		tmp = t_0;
	elseif (Float64(h / l) <= -5e-8)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * h) / Float64(Float64(l / D) * (Float64(d / M) ^ 2.0))) * -0.125)));
	elseif (Float64(h / l) <= -2e-182)
		tmp = t_0;
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	tmp = 0.0;
	if ((h / l) <= -1e+138)
		tmp = t_0;
	elseif ((h / l) <= -5e-8)
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * ((d / M) ^ 2.0))) * -0.125));
	elseif ((h / l) <= -2e-182)
		tmp = t_0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e+138], t$95$0, If[LessEqual[N[(h / l), $MachinePrecision], -5e-8], N[(w0 * N[(1.0 + N[(N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-182], t$95$0, w0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1e138 or -4.9999999999999998e-8 < (/.f64 h l) < -2.0000000000000001e-182

   1. Initial program 83.6%

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

      1. 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 54.1%

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

      1. *-commutative 54.1%

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

      2. times-frac 51.1%

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

      3. *-commutative 51.1%

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

      4. unpow2 51.1%

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

      5. unpow2 51.1%

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

      6. times-frac 69.8%

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

      7. unpow2 69.8%

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

      8. *-commutative 69.8%

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

      9. unpow2 69.8%

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

      10. associate-*l* 73.8%

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

   6. Simplified 73.8%

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

      1. unpow2 65.9%

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

   8. Applied egg-rr 73.8%

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

   if -1e138 < (/.f64 h l) < -4.9999999999999998e-8

   1. Initial program 84.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 46.0%

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

      1. associate-*r/ 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-*r/ 46.0%

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

      4. *-commutative 46.0%

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

      5. times-frac 45.9%

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

      6. unpow2 45.9%

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

      7. *-commutative 45.9%

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

      8. unpow2 45.9%

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

      9. unpow2 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in D around 0 46.0%

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

      1. *-commutative 46.0%

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

      2. times-frac 45.9%

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

      3. unpow2 45.9%

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

      4. associate-/l* 49.1%

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

      5. unpow2 49.1%

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

      6. *-commutative 49.1%

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

      7. associate-/l* 52.3%

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

      8. unpow2 52.3%

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

   9. Simplified 52.3%

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

   10. Taylor expanded in d around 0 52.3%

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

       1. unpow2 52.3%

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

       2. unpow2 52.3%

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

       3. times-frac 61.8%

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

   12. Simplified 61.8%

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

       1. frac-times 61.9%

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

       2. pow2 61.9%

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

   14. Applied egg-rr 61.9%

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

   if -2.0000000000000001e-182 < (/.f64 h l)

   1. Initial program 85.8%

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

      1. times-frac 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in M around 0 90.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot {\left(\frac{d}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 77.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0
          (sqrt
           (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))))
   (if (<= (/ h l) -1e+138)
     t_0
     (if (<= (/ h l) -5e-24)
       (*
        w0
        (sqrt (- 1.0 (* 0.25 (* (/ (* D D) l) (/ h (* (/ d M) (/ d M))))))))
       (if (<= (/ h l) -2e-182) t_0 w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = t_0;
	} else if ((h / l) <= -5e-24) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * (h / ((d / M) * (d / M)))))));
	} else if ((h / l) <= -2e-182) {
		tmp = t_0;
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)))))
    if ((h / l) <= (-1d+138)) then
        tmp = t_0
    else if ((h / l) <= (-5d-24)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d * d) / l) * (h / ((d_1 / m) * (d_1 / m)))))))
    else if ((h / l) <= (-2d-182)) then
        tmp = t_0
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = t_0;
	} else if ((h / l) <= -5e-24) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * D) / l) * (h / ((d / M) * (d / M)))))));
	} else if ((h / l) <= -2e-182) {
		tmp = t_0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	tmp = 0
	if (h / l) <= -1e+138:
		tmp = t_0
	elif (h / l) <= -5e-24:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * D) / l) * (h / ((d / M) * (d / M)))))))
	elif (h / l) <= -2e-182:
		tmp = t_0
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))))
	tmp = 0.0
	if (Float64(h / l) <= -1e+138)
		tmp = t_0;
	elseif (Float64(h / l) <= -5e-24)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * D) / l) * Float64(h / Float64(Float64(d / M) * Float64(d / M))))))));
	elseif (Float64(h / l) <= -2e-182)
		tmp = t_0;
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	tmp = 0.0;
	if ((h / l) <= -1e+138)
		tmp = t_0;
	elseif ((h / l) <= -5e-24)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * (h / ((d / M) * (d / M)))))));
	elseif ((h / l) <= -2e-182)
		tmp = t_0;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e+138], t$95$0, If[LessEqual[N[(h / l), $MachinePrecision], -5e-24], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-182], t$95$0, w0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1e138 or -4.9999999999999998e-24 < (/.f64 h l) < -2.0000000000000001e-182

   1. Initial program 82.7%

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

      1. times-frac 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in M around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. times-frac 49.4%

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

      3. *-commutative 49.4%

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

      4. unpow2 49.4%

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

      5. unpow2 49.4%

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

      6. times-frac 69.1%

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

      7. unpow2 69.1%

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

      8. *-commutative 69.1%

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

      9. unpow2 69.1%

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

      10. associate-*l* 73.4%

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

   6. Simplified 73.4%

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

      1. unpow2 65.0%

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

   8. Applied egg-rr 73.4%

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

   if -1e138 < (/.f64 h l) < -4.9999999999999998e-24

   1. Initial program 86.7%

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

      1. times-frac 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in M around 0 53.4%

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

      1. associate-*r/ 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*r/ 53.4%

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

      4. times-frac 53.4%

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

      5. unpow2 53.4%

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

      6. *-commutative 53.4%

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

      7. unpow2 53.4%

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

      8. unpow2 53.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in M around 0 53.4%

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

      1. unpow2 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-/l* 56.2%

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

      4. unpow2 56.2%

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

      5. times-frac 70.4%

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

   9. Simplified 70.4%

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

   if -2.0000000000000001e-182 < (/.f64 h l)

   1. Initial program 85.8%

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

      1. times-frac 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in M around 0 90.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D}{d}\right)}^{2}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(t_0 \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(t_0 \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (pow (/ D d) 2.0)))
   (if (<= (/ h l) -1e+138)
     (* w0 (+ 1.0 (* -0.125 (* t_0 (* M (/ (* M h) l))))))
     (if (<= (/ h l) -1e-151)
       (* w0 (+ 1.0 (* -0.125 (* (/ h (* (/ d M) (/ d M))) (/ D (/ l D))))))
       (if (<= (/ h l) -2e-182)
         (* w0 (+ 1.0 (* -0.125 (* t_0 (* M (* M (/ h l)))))))
         w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow((D / d), 2.0);
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * ((M * h) / l)))));
	} else if ((h / l) <= -1e-151) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((h / l) <= -2e-182) {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * (M * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d / d_1) ** 2.0d0
    if ((h / l) <= (-1d+138)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (t_0 * (m * ((m * h) / l)))))
    else if ((h / l) <= (-1d-151)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / ((d_1 / m) * (d_1 / m))) * (d / (l / d)))))
    else if ((h / l) <= (-2d-182)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (t_0 * (m * (m * (h / l))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow((D / d), 2.0);
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * ((M * h) / l)))));
	} else if ((h / l) <= -1e-151) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((h / l) <= -2e-182) {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * (M * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.pow((D / d), 2.0)
	tmp = 0
	if (h / l) <= -1e+138:
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * ((M * h) / l)))))
	elif (h / l) <= -1e-151:
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))))
	elif (h / l) <= -2e-182:
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * (M * (h / l))))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D / d) ^ 2.0
	tmp = 0.0
	if (Float64(h / l) <= -1e+138)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(t_0 * Float64(M * Float64(Float64(M * h) / l))))));
	elseif (Float64(h / l) <= -1e-151)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D / Float64(l / D))))));
	elseif (Float64(h / l) <= -2e-182)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(t_0 * Float64(M * Float64(M * Float64(h / l)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (D / d) ^ 2.0;
	tmp = 0.0;
	if ((h / l) <= -1e+138)
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * ((M * h) / l)))));
	elseif ((h / l) <= -1e-151)
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	elseif ((h / l) <= -2e-182)
		tmp = w0 * (1.0 + (-0.125 * (t_0 * (M * (M * (h / l))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e+138], N[(w0 * N[(1.0 + N[(-0.125 * N[(t$95$0 * N[(M * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-151], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-182], N[(w0 * N[(1.0 + N[(-0.125 * N[(t$95$0 * N[(M * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 h l) < -1e138

   1. Initial program 76.8%

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

      1. times-frac 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in M around 0 54.6%

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

      1. associate-*r/ 54.6%

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

      2. *-commutative 54.6%

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

      3. associate-*r/ 54.6%

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

      4. *-commutative 54.6%

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

      5. times-frac 54.8%

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

      6. unpow2 54.8%

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

      7. *-commutative 54.8%

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

      8. unpow2 54.8%

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

      9. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in D around 0 54.6%

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

      1. times-frac 48.7%

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

      2. unpow2 48.7%

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

      3. *-commutative 48.7%

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

      4. unpow2 48.7%

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

      5. unpow2 48.7%

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

      6. times-frac 61.0%

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

      7. unpow2 61.0%

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

      8. associate-/l* 59.0%

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

      9. associate-/r/ 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in h around 0 61.0%

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

       1. unpow2 61.0%

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

       2. *-commutative 61.0%

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

       3. associate-*r/ 59.0%

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

       4. associate-*l* 61.1%

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

   12. Simplified 61.1%

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

   13. Taylor expanded in h around 0 63.1%

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

   if -1e138 < (/.f64 h l) < -9.9999999999999994e-152

   1. Initial program 87.8%

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

      1. times-frac 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in M around 0 47.2%

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

      1. associate-*r/ 47.2%

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

      2. *-commutative 47.2%

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

      3. associate-*r/ 47.2%

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

      4. *-commutative 47.2%

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

      5. times-frac 48.5%

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

      6. unpow2 48.5%

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

      7. *-commutative 48.5%

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

      8. unpow2 48.5%

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

      9. unpow2 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in D around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. times-frac 48.5%

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

      3. unpow2 48.5%

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

      4. associate-/l* 52.9%

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

      5. unpow2 52.9%

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

      6. *-commutative 52.9%

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

      7. associate-/l* 54.6%

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

      8. unpow2 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in d around 0 54.6%

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

       1. unpow2 54.6%

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

       2. unpow2 54.6%

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

       3. times-frac 67.9%

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

   12. Simplified 67.9%

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

   if -9.9999999999999994e-152 < (/.f64 h l) < -2.0000000000000001e-182

   1. Initial program 99.7%

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

      1. times-frac 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in M around 0 40.0%

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

      1. associate-*r/ 40.0%

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

      2. *-commutative 40.0%

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

      3. associate-*r/ 40.0%

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

      4. *-commutative 40.0%

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

      5. times-frac 40.0%

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

      6. unpow2 40.0%

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

      7. *-commutative 40.0%

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

      8. unpow2 40.0%

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

      9. unpow2 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in D around 0 40.0%

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

      1. times-frac 40.0%

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

      2. unpow2 40.0%

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

      3. *-commutative 40.0%

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

      4. unpow2 40.0%

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

      5. unpow2 40.0%

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

      6. times-frac 61.5%

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

      7. unpow2 61.5%

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

      8. associate-/l* 61.5%

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

      9. associate-/r/ 61.5%

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

   9. Simplified 61.5%

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

   10. Taylor expanded in h around 0 61.5%

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

       1. unpow2 61.5%

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

       2. *-commutative 61.5%

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

       3. associate-*r/ 61.5%

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

       4. associate-*l* 82.0%

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

   12. Simplified 82.0%

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

   if -2.0000000000000001e-182 < (/.f64 h l)

   1. Initial program 85.8%

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

      1. times-frac 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in M around 0 90.5%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-182}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{M}\right)}^{2}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -50000000000:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot t_0} \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \frac{h}{t_0}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (pow (/ d M) 2.0)))
   (if (<= (/ h l) -1e+138)
     (* w0 (+ 1.0 (* -0.125 (* (pow (/ D d) 2.0) (* M (/ (* M h) l))))))
     (if (<= (/ h l) -50000000000.0)
       (* w0 (+ 1.0 (* (/ (* D h) (* (/ l D) t_0)) -0.125)))
       (if (<= (/ h l) -1e-180)
         (* w0 (+ 1.0 (* -0.125 (/ (* D (/ h t_0)) (/ l D)))))
         w0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow((d / M), 2.0);
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (pow((D / d), 2.0) * (M * ((M * h) / l)))));
	} else if ((h / l) <= -50000000000.0) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * t_0)) * -0.125));
	} else if ((h / l) <= -1e-180) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (h / t_0)) / (l / D))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / m) ** 2.0d0
    if ((h / l) <= (-1d+138)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) ** 2.0d0) * (m * ((m * h) / l)))))
    else if ((h / l) <= (-50000000000.0d0)) then
        tmp = w0 * (1.0d0 + (((d * h) / ((l / d) * t_0)) * (-0.125d0)))
    else if ((h / l) <= (-1d-180)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (h / t_0)) / (l / d))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow((d / M), 2.0);
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (Math.pow((D / d), 2.0) * (M * ((M * h) / l)))));
	} else if ((h / l) <= -50000000000.0) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * t_0)) * -0.125));
	} else if ((h / l) <= -1e-180) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (h / t_0)) / (l / D))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.pow((d / M), 2.0)
	tmp = 0
	if (h / l) <= -1e+138:
		tmp = w0 * (1.0 + (-0.125 * (math.pow((D / d), 2.0) * (M * ((M * h) / l)))))
	elif (h / l) <= -50000000000.0:
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * t_0)) * -0.125))
	elif (h / l) <= -1e-180:
		tmp = w0 * (1.0 + (-0.125 * ((D * (h / t_0)) / (l / D))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(d / M) ^ 2.0
	tmp = 0.0
	if (Float64(h / l) <= -1e+138)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(D / d) ^ 2.0) * Float64(M * Float64(Float64(M * h) / l))))));
	elseif (Float64(h / l) <= -50000000000.0)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * h) / Float64(Float64(l / D) * t_0)) * -0.125)));
	elseif (Float64(h / l) <= -1e-180)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(h / t_0)) / Float64(l / D)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (d / M) ^ 2.0;
	tmp = 0.0;
	if ((h / l) <= -1e+138)
		tmp = w0 * (1.0 + (-0.125 * (((D / d) ^ 2.0) * (M * ((M * h) / l)))));
	elseif ((h / l) <= -50000000000.0)
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * t_0)) * -0.125));
	elseif ((h / l) <= -1e-180)
		tmp = w0 * (1.0 + (-0.125 * ((D * (h / t_0)) / (l / D))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e+138], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -50000000000.0], N[(w0 * N[(1.0 + N[(N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-180], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 h l) < -1e138

   1. Initial program 76.8%

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

      1. times-frac 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in M around 0 54.6%

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

      1. associate-*r/ 54.6%

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

      2. *-commutative 54.6%

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

      3. associate-*r/ 54.6%

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

      4. *-commutative 54.6%

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

      5. times-frac 54.8%

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

      6. unpow2 54.8%

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

      7. *-commutative 54.8%

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

      8. unpow2 54.8%

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

      9. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in D around 0 54.6%

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

      1. times-frac 48.7%

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

      2. unpow2 48.7%

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

      3. *-commutative 48.7%

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

      4. unpow2 48.7%

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

      5. unpow2 48.7%

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

      6. times-frac 61.0%

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

      7. unpow2 61.0%

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

      8. associate-/l* 59.0%

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

      9. associate-/r/ 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in h around 0 61.0%

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

       1. unpow2 61.0%

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

       2. *-commutative 61.0%

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

       3. associate-*r/ 59.0%

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

       4. associate-*l* 61.1%

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

   12. Simplified 61.1%

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

   13. Taylor expanded in h around 0 63.1%

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

   if -1e138 < (/.f64 h l) < -5e10

   1. Initial program 79.5%

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

      1. times-frac 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in M around 0 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-commutative 44.4%

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

      3. associate-*r/ 44.4%

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

      4. *-commutative 44.4%

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

      5. times-frac 44.3%

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

      6. unpow2 44.3%

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

      7. *-commutative 44.3%

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

      8. unpow2 44.3%

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

      9. unpow2 44.3%

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

   6. Simplified 44.3%

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

   7. Taylor expanded in D around 0 44.4%

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

      1. *-commutative 44.4%

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

      2. times-frac 44.3%

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

      3. unpow2 44.3%

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

      4. associate-/l* 48.7%

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

      5. unpow2 48.7%

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

      6. *-commutative 48.7%

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

      7. associate-/l* 53.0%

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

      8. unpow2 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in d around 0 53.0%

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

       1. unpow2 53.0%

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

       2. unpow2 53.0%

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

       3. times-frac 60.6%

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

   12. Simplified 60.6%

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

       1. frac-times 60.7%

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

       2. pow2 60.7%

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

   14. Applied egg-rr 60.7%

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

   if -5e10 < (/.f64 h l) < -1e-180

   1. Initial program 92.7%

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

      1. times-frac 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. *-commutative 48.6%

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

      5. times-frac 50.6%

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

      6. unpow2 50.6%

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

      7. *-commutative 50.6%

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

      8. unpow2 50.6%

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

      9. unpow2 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. times-frac 50.6%

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

      3. unpow2 50.6%

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

      4. associate-/l* 54.6%

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

      5. unpow2 54.6%

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

      6. *-commutative 54.6%

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

      7. associate-/l* 54.9%

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

      8. unpow2 54.9%

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

   9. Simplified 54.9%

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

   10. Taylor expanded in d around 0 54.9%

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

       1. unpow2 54.9%

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

       2. unpow2 54.9%

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

       3. times-frac 73.8%

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

   12. Simplified 73.8%

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

       1. associate-*l/ 79.8%

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

       2. pow2 79.8%

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

   14. Applied egg-rr 79.8%

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

   if -1e-180 < (/.f64 h l)

   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 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in M around 0 89.9%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -50000000000:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot {\left(\frac{d}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot {\left(\frac{d}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e+138)
   (* w0 (+ 1.0 (* -0.125 (* (pow (/ D d) 2.0) (* M (/ (* M h) l))))))
   (if (<= (/ h l) -1e-180)
     (* w0 (+ 1.0 (* (/ (* D h) (* (/ l D) (pow (/ d M) 2.0))) -0.125)))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (pow((D / d), 2.0) * (M * ((M * h) / l)))));
	} else if ((h / l) <= -1e-180) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * pow((d / M), 2.0))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((h / l) <= (-1d+138)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) ** 2.0d0) * (m * ((m * h) / l)))))
    else if ((h / l) <= (-1d-180)) then
        tmp = w0 * (1.0d0 + (((d * h) / ((l / d) * ((d_1 / m) ** 2.0d0))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e+138) {
		tmp = w0 * (1.0 + (-0.125 * (Math.pow((D / d), 2.0) * (M * ((M * h) / l)))));
	} else if ((h / l) <= -1e-180) {
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * Math.pow((d / M), 2.0))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e+138:
		tmp = w0 * (1.0 + (-0.125 * (math.pow((D / d), 2.0) * (M * ((M * h) / l)))))
	elif (h / l) <= -1e-180:
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * math.pow((d / M), 2.0))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e+138)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(D / d) ^ 2.0) * Float64(M * Float64(Float64(M * h) / l))))));
	elseif (Float64(h / l) <= -1e-180)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * h) / Float64(Float64(l / D) * (Float64(d / M) ^ 2.0))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e+138)
		tmp = w0 * (1.0 + (-0.125 * (((D / d) ^ 2.0) * (M * ((M * h) / l)))));
	elseif ((h / l) <= -1e-180)
		tmp = w0 * (1.0 + (((D * h) / ((l / D) * ((d / M) ^ 2.0))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e+138], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-180], N[(w0 * N[(1.0 + N[(N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -1e138

   1. Initial program 76.8%

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

      1. times-frac 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in M around 0 54.6%

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

      1. associate-*r/ 54.6%

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

      2. *-commutative 54.6%

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

      3. associate-*r/ 54.6%

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

      4. *-commutative 54.6%

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

      5. times-frac 54.8%

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

      6. unpow2 54.8%

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

      7. *-commutative 54.8%

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

      8. unpow2 54.8%

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

      9. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in D around 0 54.6%

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

      1. times-frac 48.7%

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

      2. unpow2 48.7%

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

      3. *-commutative 48.7%

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

      4. unpow2 48.7%

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

      5. unpow2 48.7%

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

      6. times-frac 61.0%

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

      7. unpow2 61.0%

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

      8. associate-/l* 59.0%

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

      9. associate-/r/ 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in h around 0 61.0%

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

       1. unpow2 61.0%

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

       2. *-commutative 61.0%

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

       3. associate-*r/ 59.0%

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

       4. associate-*l* 61.1%

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

   12. Simplified 61.1%

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

   13. Taylor expanded in h around 0 63.1%

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

   if -1e138 < (/.f64 h l) < -1e-180

   1. Initial program 88.5%

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

      1. times-frac 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in M around 0 47.3%

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

      1. associate-*r/ 47.3%

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

      2. *-commutative 47.3%

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

      3. associate-*r/ 47.3%

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

      4. *-commutative 47.3%

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

      5. times-frac 48.6%

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

      6. unpow2 48.6%

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

      7. *-commutative 48.6%

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

      8. unpow2 48.6%

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

      9. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in D around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. times-frac 48.6%

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

      3. unpow2 48.6%

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

      4. associate-/l* 52.7%

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

      5. unpow2 52.7%

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

      6. *-commutative 52.7%

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

      7. associate-/l* 54.3%

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

      8. unpow2 54.3%

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

   9. Simplified 54.3%

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

   10. Taylor expanded in d around 0 54.3%

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

       1. unpow2 54.3%

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

       2. unpow2 54.3%

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

       3. times-frac 69.6%

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

   12. Simplified 69.6%

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

       1. frac-times 69.7%

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

       2. pow2 69.7%

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

   14. Applied egg-rr 69.7%

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

   if -1e-180 < (/.f64 h l)

   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 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in M around 0 89.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot h}{\frac{\ell}{D} \cdot {\left(\frac{d}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.45 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;M \leq -1.7 \cdot 10^{+23} \lor \neg \left(M \leq 4.05 \cdot 10^{-112}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M -1.45e+113)
   (* w0 (+ 1.0 (* -0.125 (* (/ h (* (/ d M) (/ d M))) (/ D (/ l D))))))
   (if (or (<= M -1.7e+23) (not (<= M 4.05e-112)))
     (* w0 (+ 1.0 (* -0.125 (* (pow (/ D d) 2.0) (* M (* M (/ h l)))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1.45e+113) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((M <= -1.7e+23) || !(M <= 4.05e-112)) {
		tmp = w0 * (1.0 + (-0.125 * (pow((D / d), 2.0) * (M * (M * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-1.45d+113)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / ((d_1 / m) * (d_1 / m))) * (d / (l / d)))))
    else if ((m <= (-1.7d+23)) .or. (.not. (m <= 4.05d-112))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) ** 2.0d0) * (m * (m * (h / l))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1.45e+113) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((M <= -1.7e+23) || !(M <= 4.05e-112)) {
		tmp = w0 * (1.0 + (-0.125 * (Math.pow((D / d), 2.0) * (M * (M * (h / l))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1.45e+113:
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))))
	elif (M <= -1.7e+23) or not (M <= 4.05e-112):
		tmp = w0 * (1.0 + (-0.125 * (math.pow((D / d), 2.0) * (M * (M * (h / l))))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1.45e+113)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D / Float64(l / D))))));
	elseif ((M <= -1.7e+23) || !(M <= 4.05e-112))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(D / d) ^ 2.0) * Float64(M * Float64(M * Float64(h / l)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -1.45e+113)
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	elseif ((M <= -1.7e+23) || ~((M <= 4.05e-112)))
		tmp = w0 * (1.0 + (-0.125 * (((D / d) ^ 2.0) * (M * (M * (h / l))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -1.45e+113], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, -1.7e+23], N[Not[LessEqual[M, 4.05e-112]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if M < -1.44999999999999992e113

   1. Initial program 76.2%

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

      1. times-frac 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 34.1%

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

      1. associate-*r/ 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*r/ 34.1%

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

      4. *-commutative 34.1%

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

      5. times-frac 34.2%

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

      6. unpow2 34.2%

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

      7. *-commutative 34.2%

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

      8. unpow2 34.2%

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

      9. unpow2 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in D around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. times-frac 34.2%

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

      3. unpow2 34.2%

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

      4. associate-/l* 39.9%

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

      5. unpow2 39.9%

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

      6. *-commutative 39.9%

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

      7. associate-/l* 42.7%

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

      8. unpow2 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in d around 0 42.7%

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

       1. unpow2 42.7%

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

       2. unpow2 42.7%

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

       3. times-frac 63.0%

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

   12. Simplified 63.0%

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

   if -1.44999999999999992e113 < M < -1.69999999999999996e23 or 4.05000000000000011e-112 < M

   1. Initial program 82.4%

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

      1. times-frac 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 45.8%

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

      1. associate-*r/ 45.8%

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

      2. *-commutative 45.8%

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

      3. associate-*r/ 45.8%

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

      4. *-commutative 45.8%

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

      5. times-frac 48.8%

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

      6. unpow2 48.8%

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

      7. *-commutative 48.8%

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

      8. unpow2 48.8%

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

      9. unpow2 48.8%

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

   6. Simplified 48.8%

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

   7. Taylor expanded in D around 0 45.8%

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

      1. times-frac 47.8%

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

      2. unpow2 47.8%

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

      3. *-commutative 47.8%

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

      4. unpow2 47.8%

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

      5. unpow2 47.8%

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

      6. times-frac 58.6%

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

      7. unpow2 58.6%

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

      8. associate-/l* 59.5%

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

      9. associate-/r/ 61.4%

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

   9. Simplified 61.4%

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

   10. Taylor expanded in h around 0 58.6%

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

       1. unpow2 58.6%

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

       2. *-commutative 58.6%

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

       3. associate-*r/ 61.4%

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

       4. associate-*l* 70.1%

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

   12. Simplified 70.1%

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

   if -1.69999999999999996e23 < M < 4.05000000000000011e-112

   1. Initial program 89.7%

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

      1. times-frac 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in M around 0 88.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.45 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;M \leq -1.7 \cdot 10^{+23} \lor \neg \left(M \leq 4.05 \cdot 10^{-112}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 71.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.75 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;M \leq -2.4 \cdot 10^{+21} \lor \neg \left(M \leq 4.7 \cdot 10^{-119}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M -1.75e+113)
   (* w0 (+ 1.0 (* -0.125 (* (/ h (* (/ d M) (/ d M))) (/ D (/ l D))))))
   (if (or (<= M -2.4e+21) (not (<= M 4.7e-119)))
     (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (* (/ h l) (* M M))))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1.75e+113) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((M <= -2.4e+21) || !(M <= 4.7e-119)) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-1.75d+113)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / ((d_1 / m) * (d_1 / m))) * (d / (l / d)))))
    else if ((m <= (-2.4d+21)) .or. (.not. (m <= 4.7d-119))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((h / l) * (m * m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1.75e+113) {
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	} else if ((M <= -2.4e+21) || !(M <= 4.7e-119)) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1.75e+113:
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))))
	elif (M <= -2.4e+21) or not (M <= 4.7e-119):
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1.75e+113)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / Float64(Float64(d / M) * Float64(d / M))) * Float64(D / Float64(l / D))))));
	elseif ((M <= -2.4e+21) || !(M <= 4.7e-119))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h / l) * Float64(M * M))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -1.75e+113)
		tmp = w0 * (1.0 + (-0.125 * ((h / ((d / M) * (d / M))) * (D / (l / D)))));
	elseif ((M <= -2.4e+21) || ~((M <= 4.7e-119)))
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -1.75e+113], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, -2.4e+21], N[Not[LessEqual[M, 4.7e-119]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if M < -1.75e113

   1. Initial program 76.2%

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

      1. times-frac 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 34.1%

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

      1. associate-*r/ 34.1%

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

      2. *-commutative 34.1%

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

      3. associate-*r/ 34.1%

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

      4. *-commutative 34.1%

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

      5. times-frac 34.2%

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

      6. unpow2 34.2%

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

      7. *-commutative 34.2%

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

      8. unpow2 34.2%

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

      9. unpow2 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in D around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. times-frac 34.2%

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

      3. unpow2 34.2%

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

      4. associate-/l* 39.9%

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

      5. unpow2 39.9%

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

      6. *-commutative 39.9%

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

      7. associate-/l* 42.7%

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

      8. unpow2 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in d around 0 42.7%

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

       1. unpow2 42.7%

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

       2. unpow2 42.7%

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

       3. times-frac 63.0%

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

   12. Simplified 63.0%

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

   if -1.75e113 < M < -2.4e21 or 4.70000000000000002e-119 < M

   1. Initial program 82.6%

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

      1. 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 45.4%

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

      1. associate-*r/ 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*r/ 45.4%

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

      4. *-commutative 45.4%

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

      5. times-frac 48.3%

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

      6. unpow2 48.3%

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

      7. *-commutative 48.3%

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

      8. unpow2 48.3%

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

      9. unpow2 48.3%

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

   6. Simplified 48.3%

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

   7. Taylor expanded in D around 0 45.4%

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

      1. times-frac 47.4%

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

      2. unpow2 47.4%

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

      3. *-commutative 47.4%

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

      4. unpow2 47.4%

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

      5. unpow2 47.4%

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

      6. times-frac 59.0%

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

      7. unpow2 59.0%

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

      8. associate-/l* 59.8%

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

      9. associate-/r/ 61.7%

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

   9. Simplified 61.7%

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

       1. unpow2 61.7%

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

   11. Applied egg-rr 61.7%

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

   if -2.4e21 < M < 4.70000000000000002e-119

   1. Initial program 89.6%

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

      1. times-frac 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in M around 0 88.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.75 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\frac{d}{M} \cdot \frac{d}{M}} \cdot \frac{D}{\frac{\ell}{D}}\right)\right)\\ \mathbf{elif}\;M \leq -2.4 \cdot 10^{+21} \lor \neg \left(M \leq 4.7 \cdot 10^{-119}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 11: 73.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -5e-133)
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (* (/ h l) (* M M))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-133) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((h / l) <= (-5d-133)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((h / l) * (m * m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-133) {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e-133:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-133)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h / l) * Float64(M * M))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-133)
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-133], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.9999999999999999e-133

   1. Initial program 83.2%

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

      1. times-frac 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. associate-*r/ 51.1%

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

      2. *-commutative 51.1%

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

      3. associate-*r/ 51.1%

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

      4. *-commutative 51.1%

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

      5. times-frac 52.1%

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

      6. unpow2 52.1%

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

      7. *-commutative 52.1%

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

      8. unpow2 52.1%

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

      9. unpow2 52.1%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in D around 0 51.1%

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

      1. times-frac 49.4%

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

      2. unpow2 49.4%

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

      3. *-commutative 49.4%

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

      4. unpow2 49.4%

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

      5. unpow2 49.4%

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

      6. times-frac 63.1%

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

      7. unpow2 63.1%

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

      8. associate-/l* 60.5%

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

      9. associate-/r/ 62.4%

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

   9. Simplified 62.4%

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

       1. unpow2 62.4%

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

   11. Applied egg-rr 62.4%

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

   if -4.9999999999999999e-133 < (/.f64 h l)

   1. Initial program 86.2%

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

      1. times-frac 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in M around 0 88.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 12: 67.8% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

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
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 84.8%

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

   1. times-frac 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in M around 0 68.9%

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

5. Final simplification 68.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023195 
(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))))))