Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.3%

Time: 15.4s

Alternatives: 3

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: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left({\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -5e-306)
   (* w0 (sqrt (- 1.0 (* h (* (pow (* (* D M) (/ 0.5 d)) 2.0) (/ 1.0 l))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-306) {
		tmp = w0 * sqrt((1.0 - (h * (pow(((D * M) * (0.5 / d)), 2.0) * (1.0 / 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 ((h / l) <= (-5d-306)) then
        tmp = w0 * sqrt((1.0d0 - (h * ((((d * m) * (0.5d0 / d_1)) ** 2.0d0) * (1.0d0 / 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 ((h / l) <= -5e-306) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow(((D * M) * (0.5 / d)), 2.0) * (1.0 / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e-306:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow(((D * M) * (0.5 / d)), 2.0) * (1.0 / l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0) * Float64(1.0 / l))))));
	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-306)
		tmp = w0 * sqrt((1.0 - (h * ((((D * M) * (0.5 / d)) ^ 2.0) * (1.0 / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-306], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99999999999999998e-306

   1. Initial program 72.9%

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

   3. Simplified 72.2%

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

      1. clear-num 72.2%

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

      2. un-div-inv 72.9%

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

      3. *-commutative 72.9%

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

      4. associate-*l/ 74.1%

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

      5. associate-*r/ 71.9%

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

      6. div-inv 71.9%

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

      7. metadata-eval 71.9%

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

   6. Applied egg-rr 71.9%

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

      1. associate-/r/ 81.1%

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

      2. associate-*r/ 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-/l* 82.0%

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

      5. associate-*r/ 82.0%

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

   8. Simplified 82.0%

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

      1. div-inv 82.0%

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

      2. associate-*r* 83.9%

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

   10. Applied egg-rr 83.9%

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

   if -4.99999999999999998e-306 < (/.f64 h l)

   1. Initial program 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in D around 0 96.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left({\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot 0.5\right) \cdot \frac{D}{d}\\ \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(t\_0 \cdot t\_0\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (* M 0.5) (/ D d))))
   (if (<= (* (/ h l) (pow (/ (* D M) (* d 2.0)) 2.0)) -1e-13)
     (* w0 (sqrt (- 1.0 (/ (* h (* t_0 t_0)) l))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * 0.5) * (D / d);
	double tmp;
	if (((h / l) * pow(((D * M) / (d * 2.0)), 2.0)) <= -1e-13) {
		tmp = w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / 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 = (m * 0.5d0) * (d / d_1)
    if (((h / l) * (((d * m) / (d_1 * 2.0d0)) ** 2.0d0)) <= (-1d-13)) then
        tmp = w0 * sqrt((1.0d0 - ((h * (t_0 * t_0)) / 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 = (M * 0.5) * (D / d);
	double tmp;
	if (((h / l) * Math.pow(((D * M) / (d * 2.0)), 2.0)) <= -1e-13) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * 0.5) * (D / d)
	tmp = 0
	if ((h / l) * math.pow(((D * M) / (d * 2.0)), 2.0)) <= -1e-13:
		tmp = w0 * math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * 0.5) * Float64(D / d))
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)) <= -1e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_0 * t_0)) / l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * 0.5) * (D / d);
	tmp = 0.0;
	if (((h / l) * (((D * M) / (d * 2.0)) ^ 2.0)) <= -1e-13)
		tmp = w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e-13], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $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)) < -1e-13

   1. Initial program 59.1%

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

   3. Simplified 57.9%

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

      1. associate-*r/ 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l/ 67.4%

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

      4. associate-*r/ 64.7%

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

      5. div-inv 64.7%

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

      6. metadata-eval 64.7%

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

   6. Applied egg-rr 64.7%

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

      1. unpow2 64.7%

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

   8. Applied egg-rr 64.7%

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

   if -1e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in D around 0 97.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% 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 79.4%

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

3. Simplified 79.0%

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

5. Taylor expanded in D around 0 69.1%

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

6. Final simplification 69.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024058 
(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))))))