Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.9%

Time: 17.6s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 0.04:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l))))
   (if (<= t_0 0.04) (* 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(((D * M) / (d * 2.0)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.04) {
		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 = (((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)
    if (t_0 <= 0.04d0) 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(((D * M) / (d * 2.0)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.04) {
		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(((D * M) / (d * 2.0)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 0.04:
		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(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 0.04)
		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 = (((D * M) / (d * 2.0)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 0.04)
		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[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.04], 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)) < 0.0400000000000000008

   1. Initial program 91.2%

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

   if 0.0400000000000000008 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in D around 0 60.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.04:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-7}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* D M) 1e-7)
   w0
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* 0.5 (* M (/ D d))) 2.0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 1e-7) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((0.5 * (M * (D / d))), 2.0))));
	}
	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 * m) <= 1d-7) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((0.5d0 * (m * (d / d_1))) ** 2.0d0))))
    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 * M) <= 1e-7) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((0.5 * (M * (D / d))), 2.0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 1e-7:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((0.5 * (M * (D / d))), 2.0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 1e-7)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 1e-7)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * ((0.5 * (M * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 1e-7], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999995e-8

   1. Initial program 87.1%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in D around 0 84.3%

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

   if 9.9999999999999995e-8 < (*.f64 M D)

   1. Initial program 62.1%

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

   3. Simplified 65.2%

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

      1. associate-*r/ 68.7%

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

      2. add-sqr-sqrt 68.7%

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

      3. pow2 68.7%

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

      4. unpow2 68.7%

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

      5. sqrt-prod 39.6%

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

      6. add-sqr-sqrt 68.7%

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

      7. div-inv 68.7%

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

      8. metadata-eval 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. associate-*r/ 65.2%

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

      2. expm1-log1p-u 28.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

      3. expm1-udef 28.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

      4. log1p-udef 28.1%

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

      5. add-exp-log 65.2%

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

      6. expm1-log1p-u 64.1%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\left(1 + {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) - 1\right)}\right)\right)} \]

      7. expm1-udef 64.1%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(1 + {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) - 1\right)}\right)} - 1\right)} \]

   8. Applied egg-rr 61.0%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{\left(D \cdot 0.5\right) \cdot M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 61.0%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{\left(D \cdot 0.5\right) \cdot M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      2. expm1-log1p 62.1%

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

      3. *-commutative 62.1%

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

      4. *-commutative 62.1%

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

      5. associate-*r* 62.1%

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

      6. associate-*r/ 62.1%

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

      7. *-commutative 62.1%

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

      8. associate-*r/ 65.2%

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

   10. Simplified 65.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-7}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(D * 0.5) * Float64(M / d)) ^ 2.0) * h) / l))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (((((D * 0.5) * (M / 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[(N[(D * 0.5), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

3. Simplified 81.2%

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

   1. associate-*r/ 86.9%

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

   2. add-sqr-sqrt 86.9%

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

   3. pow2 86.9%

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

   4. unpow2 86.9%

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

   5. sqrt-prod 60.7%

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

   6. add-sqr-sqrt 86.9%

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

   7. div-inv 86.9%

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

   8. metadata-eval 86.9%

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

6. Applied egg-rr 86.9%

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

7. Final simplification 86.9%

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

Alternative 4: 67.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(w0 \cdot h\right) \cdot \frac{{M}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3e-46)
   w0
   (+ w0 (* -0.125 (* (* (/ D d) (/ D d)) (* (* w0 h) (/ (pow M 2.0) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3e-46) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((w0 * h) * (pow(M, 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 <= 3d-46) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * h) * ((m ** 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 <= 3e-46) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((w0 * h) * (Math.pow(M, 2.0) / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3e-46:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((w0 * h) * (math.pow(M, 2.0) / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3e-46)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * h) * Float64((M ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3e-46)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((w0 * h) * ((M ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3e-46], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] * N[(N[Power[M, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.99999999999999987e-46

   1. Initial program 84.2%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in D around 0 77.2%

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

   if 2.99999999999999987e-46 < M

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in D around 0 41.5%

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

      1. times-frac 42.9%

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

   7. Simplified 42.9%

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

      1. add-sqr-sqrt 42.9%

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

      2. sqrt-div 42.9%

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

      3. unpow2 42.9%

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

      4. sqrt-prod 26.3%

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

      5. add-sqr-sqrt 36.0%

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

      6. unpow2 36.0%

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

      7. sqrt-prod 19.4%

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

      8. add-sqr-sqrt 37.4%

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

      9. sqrt-div 37.4%

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

      10. unpow2 37.4%

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

      11. sqrt-prod 27.6%

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

      12. add-sqr-sqrt 41.0%

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

      13. unpow2 41.0%

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

      14. sqrt-prod 28.0%

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

      15. add-sqr-sqrt 51.3%

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

   9. Applied egg-rr 51.3%

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

       1. expm1-log1p-u 41.4%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)\right)}\right) \]

       2. expm1-udef 41.3%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} - 1\right)}\right) \]

       3. associate-/l* 41.2%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}}\right)} - 1\right)\right) \]

   11. Applied egg-rr 41.2%

       \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 41.2%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}\right)\right)}\right) \]

       2. expm1-log1p 51.0%

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

       3. associate-/r/ 51.3%

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

   13. Simplified 51.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(w0 \cdot h\right) \cdot \frac{{M}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{{M}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.8e-45)
   w0
   (+ w0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* (pow M 2.0) (* w0 h)) l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.8e-45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((pow(M, 2.0) * (w0 * h)) / 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 <= 1.8d-45) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (((m ** 2.0d0) * (w0 * h)) / 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 <= 1.8e-45) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((Math.pow(M, 2.0) * (w0 * h)) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.8e-45:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * ((math.pow(M, 2.0) * (w0 * h)) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.8e-45)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64((M ^ 2.0) * Float64(w0 * h)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.8e-45)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * (((M ^ 2.0) * (w0 * h)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.8e-45], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[M, 2.0], $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8e-45

   1. Initial program 84.2%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in D around 0 77.2%

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

   if 1.8e-45 < M

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in D around 0 41.5%

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

      1. times-frac 42.9%

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

   7. Simplified 42.9%

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

      1. add-sqr-sqrt 42.9%

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

      2. sqrt-div 42.9%

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

      3. unpow2 42.9%

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

      4. sqrt-prod 26.3%

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

      5. add-sqr-sqrt 36.0%

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

      6. unpow2 36.0%

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

      7. sqrt-prod 19.4%

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

      8. add-sqr-sqrt 37.4%

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

      9. sqrt-div 37.4%

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

      10. unpow2 37.4%

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

      11. sqrt-prod 27.6%

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

      12. add-sqr-sqrt 41.0%

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

      13. unpow2 41.0%

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

      14. sqrt-prod 28.0%

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

      15. add-sqr-sqrt 51.3%

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

   9. Applied egg-rr 51.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{{M}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.06 \cdot 10^{-48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{w0 \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.06e-48)
   w0
   (+ w0 (* -0.125 (/ (pow (/ M (/ d D)) 2.0) (/ l (* w0 h)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.06e-48) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((M / (d / D)), 2.0) / (l / (w0 * h))));
	}
	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 <= 1.06d-48) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((m / (d_1 / d)) ** 2.0d0) / (l / (w0 * h))))
    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 <= 1.06e-48) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((M / (d / D)), 2.0) / (l / (w0 * h))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.06e-48:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((M / (d / D)), 2.0) / (l / (w0 * h))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.06e-48)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(M / Float64(d / D)) ^ 2.0) / Float64(l / Float64(w0 * h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.06e-48)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((M / (d / D)) ^ 2.0) / (l / (w0 * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.06e-48], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.0600000000000001e-48

   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 84.6%

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

   5. Taylor expanded in D around 0 77.5%

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

   if 1.0600000000000001e-48 < M

   1. Initial program 70.8%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in D around 0 40.9%

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

      1. times-frac 42.2%

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

   7. Simplified 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-div 42.2%

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

      3. unpow2 42.2%

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

      4. sqrt-prod 25.8%

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

      5. add-sqr-sqrt 35.5%

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

      6. unpow2 35.5%

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

      7. sqrt-prod 19.1%

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

      8. add-sqr-sqrt 36.8%

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

      9. sqrt-div 36.8%

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

      10. unpow2 36.8%

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

      11. sqrt-prod 27.2%

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

      12. add-sqr-sqrt 40.3%

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

      13. unpow2 40.3%

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

      14. sqrt-prod 27.6%

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

      15. add-sqr-sqrt 50.5%

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

   9. Applied egg-rr 50.5%

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

       1. expm1-log1p-u 40.7%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)\right)}\right) \]

       2. expm1-udef 40.7%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{\ell}\right)} - 1\right)}\right) \]

       3. associate-/l* 40.5%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}}\right)} - 1\right)\right) \]

   11. Applied egg-rr 40.5%

       \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}\right)} - 1\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-def 40.5%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{M}^{2}}{\frac{\ell}{h \cdot w0}}\right)\right)}\right) \]

       2. expm1-log1p 50.2%

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

       3. associate-/r/ 50.5%

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

   13. Simplified 50.5%

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

   14. Taylor expanded in D around 0 40.9%

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

       1. times-frac 42.2%

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

       2. unpow2 42.2%

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

       3. unpow2 42.2%

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

       4. times-frac 50.5%

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

       5. unpow2 50.5%

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

       6. associate-/l* 50.2%

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

       7. associate-*r/ 50.3%

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

       8. unpow2 50.3%

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

       9. unpow2 50.3%

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

       10. swap-sqr 61.9%

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

       11. *-commutative 61.9%

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

       12. *-commutative 61.9%

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

       13. unpow1 61.9%

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

       14. pow-plus 61.9%

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

       15. associate-*r/ 60.4%

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

       16. associate-/l* 61.9%

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

       17. metadata-eval 61.9%

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

   16. Simplified 61.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.06 \cdot 10^{-48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{w0 \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% 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 80.9%

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

3. Simplified 81.2%

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

5. Taylor expanded in D around 0 71.2%

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

6. Final simplification 71.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))