Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 86.1%

Time: 17.4s

Alternatives: 5

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow(((D / d) * (M * 0.5)), 2.0)) / 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 - ((h * (((d / d_1) * (m * 0.5d0)) ** 2.0d0)) / 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 - ((h * Math.pow(((D / d) * (M * 0.5)), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*l* 84.4%

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

   3. associate-/r* 84.4%

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

   4. associate-*r/ 83.3%

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

   5. associate-*l/ 84.1%

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

   6. *-commutative 84.1%

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

   7. associate-/r* 84.1%

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

   8. associate-*r/ 83.0%

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

   9. associate-*l/ 84.1%

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

   10. *-commutative 84.1%

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

   11. associate-*l* 82.9%

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

   12. unpow2 82.9%

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

   13. *-commutative 82.9%

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

   14. associate-*l/ 88.6%

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

   15. *-commutative 88.6%

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

   16. associate-*l/ 87.3%

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

6. Applied egg-rr 87.3%

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

   1. *-commutative 87.3%

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

   2. times-frac 88.6%

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

   3. div-inv 88.6%

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

   4. metadata-eval 88.6%

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

8. Applied egg-rr 88.6%

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

9. Final simplification 88.6%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((D * (M / (d * 2.0))), 2.0) / 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 - (h * (((d * (m / (d_1 * 2.0d0))) ** 2.0d0) / 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 - (h * (Math.pow((D * (M / (d * 2.0))), 2.0) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*l* 84.4%

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

   3. associate-/r* 84.4%

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

   4. associate-*r/ 83.3%

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

   5. associate-*l/ 84.1%

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

   6. *-commutative 84.1%

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

   7. associate-/r* 84.1%

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

   8. associate-*r/ 83.0%

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

   9. associate-*l/ 84.1%

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

   10. *-commutative 84.1%

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

   11. associate-*l* 82.9%

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

   12. unpow2 82.9%

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

   13. expm1-log1p-u 59.8%

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

   14. expm1-undefine 59.8%

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

6. Applied egg-rr 81.8%

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

   1. associate--l+ 81.8%

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

   2. metadata-eval 81.8%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 81.8%

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

   4. associate-*l/ 87.3%

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

   5. associate-/l* 86.5%

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

   6. associate-/l* 87.4%

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

8. Simplified 87.4%

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

9. Final simplification 87.4%

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

Alternative 3: 78.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-283}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{1}{\frac{\ell}{h} \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -1e-283)
   (* w0 (+ 1.0 (* (/ 1.0 (* (/ l h) (pow (/ (/ d D) M) 2.0))) -0.125)))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-283) {
		tmp = w0 * (1.0 + ((1.0 / ((l / h) * pow(((d / D) / M), 2.0))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-1d-283)) then
        tmp = w0 * (1.0d0 + ((1.0d0 / ((l / h) * (((d_1 / d) / m) ** 2.0d0))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-283) {
		tmp = w0 * (1.0 + ((1.0 / ((l / h) * Math.pow(((d / D) / M), 2.0))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-283:
		tmp = w0 * (1.0 + ((1.0 / ((l / h) * math.pow(((d / D) / M), 2.0))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-283)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(1.0 / Float64(Float64(l / h) * (Float64(Float64(d / D) / M) ^ 2.0))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-283)
		tmp = w0 * (1.0 + ((1.0 / ((l / h) * (((d / D) / M) ^ 2.0))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-283], N[(w0 * N[(1.0 + N[(N[(1.0 / N[(N[(l / h), $MachinePrecision] * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.99999999999999947e-284

   1. Initial program 79.6%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in D around 0 54.9%

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

      1. *-commutative 54.9%

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

   7. Simplified 54.9%

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

      1. add-sqr-sqrt 37.8%

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

      2. times-frac 38.8%

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

      3. sqrt-prod 37.4%

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

      4. sqrt-pow1 30.9%

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

      5. metadata-eval 30.9%

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

      6. pow1 30.9%

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

      7. sqrt-prod 24.0%

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

      8. sqrt-pow1 22.8%

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

      9. metadata-eval 22.8%

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

      10. pow1 22.8%

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

   9. Applied egg-rr 38.3%

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

   10. Taylor expanded in D around 0 54.9%

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

       1. associate-*r* 57.0%

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

       2. unpow2 57.0%

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

       3. unpow2 57.0%

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

       4. swap-sqr 69.5%

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

       5. unpow2 69.5%

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

   12. Simplified 69.5%

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

       1. clear-num 69.5%

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

       2. inv-pow 69.5%

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

       3. times-frac 69.3%

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

       4. unpow2 69.3%

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

       5. unpow2 69.3%

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

       6. frac-times 72.3%

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

       7. pow2 72.3%

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

   14. Applied egg-rr 72.3%

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

       1. unpow-1 72.3%

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

       2. *-commutative 72.3%

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

       3. associate-/r* 72.3%

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

   16. Simplified 72.3%

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

   if -9.99999999999999947e-284 < (/.f64 h l)

   1. Initial program 84.6%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in D around 0 95.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-283}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{1}{\frac{\ell}{h} \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{2}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2000000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \frac{-0.125}{\frac{\ell}{h} \cdot {\left(\frac{d}{D \cdot M}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2000000000000.0)
   w0
   (+ w0 (* w0 (/ -0.125 (* (/ l h) (pow (/ d (* D M)) 2.0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2000000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 / ((l / h) * pow((d / (D * M)), 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 <= 2000000000000.0d0) then
        tmp = w0
    else
        tmp = w0 + (w0 * ((-0.125d0) / ((l / h) * ((d_1 / (d * m)) ** 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 <= 2000000000000.0) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 / ((l / h) * Math.pow((d / (D * M)), 2.0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 2000000000000.0:
		tmp = w0
	else:
		tmp = w0 + (w0 * (-0.125 / ((l / h) * math.pow((d / (D * M)), 2.0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2000000000000.0)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(-0.125 / Float64(Float64(l / h) * (Float64(d / Float64(D * M)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 2000000000000.0)
		tmp = w0;
	else
		tmp = w0 + (w0 * (-0.125 / ((l / h) * ((d / (D * M)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2000000000000.0], w0, N[(w0 + N[(w0 * N[(-0.125 / N[(N[(l / h), $MachinePrecision] * N[Power[N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2e12

   1. Initial program 82.0%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in D around 0 70.6%

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

   if 2e12 < D

   1. Initial program 81.0%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in D around 0 34.5%

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

      1. *-commutative 34.5%

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

   7. Simplified 34.5%

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

      1. add-sqr-sqrt 25.3%

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

      2. times-frac 27.4%

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

      3. sqrt-prod 27.4%

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

      4. sqrt-pow1 27.4%

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

      5. metadata-eval 27.4%

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

      6. pow1 27.4%

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

      7. sqrt-prod 24.1%

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

      8. sqrt-pow1 15.7%

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

      9. metadata-eval 15.7%

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

      10. pow1 15.7%

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

   9. Applied egg-rr 52.1%

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

   10. Taylor expanded in D around 0 34.5%

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

       1. associate-*r* 34.5%

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

       2. unpow2 34.5%

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

       3. unpow2 34.5%

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

       4. swap-sqr 62.5%

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

       5. unpow2 62.5%

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

   12. Simplified 62.5%

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

       1. +-commutative 62.5%

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

       2. distribute-rgt-in 62.5%

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

       3. *-commutative 62.5%

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

       4. clear-num 62.5%

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

       5. un-div-inv 62.5%

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

       6. times-frac 63.6%

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

       7. unpow2 63.6%

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

       8. unpow2 63.6%

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

       9. frac-times 72.1%

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

       10. pow2 72.1%

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

       11. *-un-lft-identity 72.1%

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

   14. Applied egg-rr 72.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2000000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \frac{-0.125}{\frac{\ell}{h} \cdot {\left(\frac{d}{D \cdot M}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% 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 81.8%

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

3. Simplified 82.9%

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

5. Taylor expanded in D around 0 66.4%

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

6. Final simplification 66.4%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))))