Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 87.1%

Time: 19.2s

Alternatives: 6

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: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 0.01) (* 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) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.01) {
		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) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 0.01d0) 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) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.01) {
		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) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 0.01:
		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(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 0.01)
		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) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 0.01)
		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[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], 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 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0100000000000000002

   1. Initial program 85.1%

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

   if 0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in D around 0 67.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e-313)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D (/ (/ M 2.0) d)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-313) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D * ((M / 2.0) / 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) <= (-2d-313)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d * ((m / 2.0d0) / 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) <= -2e-313) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D * ((M / 2.0) / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e-313:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D * ((M / 2.0) / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-313)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D * Float64(Float64(M / 2.0) / 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) <= -2e-313)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D * ((M / 2.0) / 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], -2e-313], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999998e-313

   1. Initial program 74.4%

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

   3. Simplified 72.5%

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

   if -1.99999999998e-313 < (/.f64 h l)

   1. Initial program 84.1%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in D around 0 92.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e-313)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (* 2.0 (/ d M))) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e-313) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / (2.0 * (d / M))), 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) <= (-2d-313)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / (2.0d0 * (d_1 / m))) ** 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) <= -2e-313) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / (2.0 * (d / M))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e-313:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / (2.0 * (d / M))), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e-313)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(2.0 * Float64(d / M))) ^ 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) <= -2e-313)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / (2.0 * (d / M))) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e-313], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(2.0 * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999998e-313

   1. Initial program 74.4%

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

   3. Simplified 72.5%

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

      1. clear-num 72.5%

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

      2. un-div-inv 72.8%

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

      3. associate-/r/ 72.7%

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

   6. Applied egg-rr 72.7%

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

   if -1.99999999998e-313 < (/.f64 h l)

   1. Initial program 84.1%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in D around 0 92.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3.3 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 3.3e+143)
   w0
   (+ w0 (* w0 (* (* (/ h l) (pow (* D (/ M d)) 2.0)) -0.125)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.3e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((h / l) * pow((D * (M / d)), 2.0)) * -0.125));
	}
	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 (d <= 3.3d+143) then
        tmp = w0
    else
        tmp = w0 + (w0 * (((h / l) * ((d * (m / d_1)) ** 2.0d0)) * (-0.125d0)))
    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 (D <= 3.3e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((h / l) * Math.pow((D * (M / d)), 2.0)) * -0.125));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 3.3e+143:
		tmp = w0
	else:
		tmp = w0 + (w0 * (((h / l) * math.pow((D * (M / d)), 2.0)) * -0.125))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 3.3e+143)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(Float64(Float64(h / l) * (Float64(D * Float64(M / d)) ^ 2.0)) * -0.125)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 3.3e+143)
		tmp = w0;
	else
		tmp = w0 + (w0 * (((h / l) * ((D * (M / d)) ^ 2.0)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 3.3e+143], w0, N[(w0 + N[(w0 * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.3e143

   1. Initial program 77.7%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in D around 0 68.0%

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

   if 3.3e143 < D

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

      1. unpow2 85.1%

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

      2. unpow2 85.1%

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

      3. associate-/r* 85.1%

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

      4. associate-*r/ 85.1%

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

      5. *-commutative 85.1%

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

      6. associate-*r/ 89.4%

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

      7. *-commutative 89.4%

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

      8. associate-*l/ 89.4%

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

      9. associate-/r* 89.4%

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

      10. clear-num 89.4%

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

      11. associate-*l/ 89.4%

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

      12. *-un-lft-identity 89.4%

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

      13. associate-/r/ 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. associate-*l/ 85.1%

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

      2. *-commutative 85.1%

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

      3. associate-*l/ 85.1%

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

   8. Applied egg-rr 85.1%

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

   9. Taylor expanded in D around 0 16.1%

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

       1. +-commutative 16.1%

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

       2. *-commutative 16.1%

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

       3. fma-define 16.1%

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

       4. associate-*r* 16.2%

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

       5. unpow2 16.2%

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

       6. unpow2 16.2%

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

       7. swap-sqr 54.7%

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

       8. unpow2 54.7%

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

       9. times-frac 54.2%

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

       10. unpow2 54.2%

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

       11. unpow2 54.2%

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

       12. times-frac 72.9%

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

       13. unpow2 72.9%

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

       14. associate-/l* 72.9%

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

   11. Simplified 72.9%

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

       1. fma-undefine 72.9%

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

       2. distribute-rgt-in 72.9%

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

       3. *-un-lft-identity 72.9%

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

   13. Applied egg-rr 72.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.3 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+106}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e+106) w0 (* w0 (* -0.125 (* h (/ (pow (* D (/ M d)) 2.0) l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+106) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 * (h * (pow((D * (M / d)), 2.0) / l)));
	}
	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 <= 9d+106) then
        tmp = w0
    else
        tmp = w0 * ((-0.125d0) * (h * (((d * (m / d_1)) ** 2.0d0) / l)))
    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 <= 9e+106) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 * (h * (Math.pow((D * (M / d)), 2.0) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e+106:
		tmp = w0
	else:
		tmp = w0 * (-0.125 * (h * (math.pow((D * (M / d)), 2.0) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e+106)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(-0.125 * Float64(h * Float64((Float64(D * Float64(M / d)) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e+106)
		tmp = w0;
	else
		tmp = w0 * (-0.125 * (h * (((D * (M / d)) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e+106], w0, N[(w0 * N[(-0.125 * N[(h * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999994e106

   1. Initial program 79.3%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in D around 0 70.1%

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

   if 8.9999999999999994e106 < M

   1. Initial program 72.5%

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

   3. Simplified 75.7%

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

      1. unpow2 75.7%

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

      2. unpow2 75.7%

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

      3. associate-/r* 75.7%

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

      4. associate-*r/ 72.5%

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

      5. *-commutative 72.5%

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

      6. associate-*r/ 67.3%

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

      7. *-commutative 67.3%

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

      8. associate-*l/ 70.4%

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

      9. associate-/r* 70.4%

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

      10. clear-num 70.4%

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

      11. associate-*l/ 70.4%

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

      12. *-un-lft-identity 70.4%

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

      13. associate-/r/ 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-*l/ 75.7%

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

   8. Applied egg-rr 75.7%

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

   9. Taylor expanded in D around 0 35.6%

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

       1. +-commutative 35.6%

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

       2. *-commutative 35.6%

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

       3. fma-define 35.6%

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

       4. associate-*r* 43.5%

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

       5. unpow2 43.5%

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

       6. unpow2 43.5%

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

       7. swap-sqr 56.3%

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

       8. unpow2 56.3%

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

       9. times-frac 56.0%

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

       10. unpow2 56.0%

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

       11. unpow2 56.0%

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

       12. times-frac 66.6%

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

       13. unpow2 66.6%

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

       14. associate-/l* 69.8%

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

   11. Simplified 69.8%

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

   12. Taylor expanded in D around inf 29.4%

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

   14. Simplified 34.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+106}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% 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 78.4%

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

3. Simplified 76.9%

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

5. Taylor expanded in D around 0 66.2%

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

6. Final simplification 66.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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))))))