Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 87.5%

Time: 29.3s

Alternatives: 6

Speedup: 1.8×

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: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((((M * (0.5 / d)) * D) / l) * (((D * (M * 0.5)) / d) * h))));
}

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 * (0.5d0 / d_1)) * d) / l) * (((d * (m * 0.5d0)) / d_1) * h))))
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 - ((((M * (0.5 / d)) * D) / l) * (((D * (M * 0.5)) / d) * h))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 83.2%

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

   1. clear-num 82.8%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-*l/ 83.6%

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

   5. associate-*r/ 83.2%

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

   6. div-inv 83.2%

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

   7. metadata-eval 83.2%

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

6. Applied egg-rr 83.2%

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

   1. *-commutative 83.2%

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

   2. associate-*l/ 83.6%

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

   3. associate-/l* 83.2%

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

   4. associate-*r/ 83.2%

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

8. Simplified 83.2%

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

   1. unpow2 83.2%

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

   2. div-inv 83.2%

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

   3. times-frac 91.4%

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

   4. *-commutative 91.4%

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

   5. associate-*l* 88.8%

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

   6. *-commutative 88.8%

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

   7. associate-*l* 89.2%

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

10. Applied egg-rr 89.2%

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

    1. associate-*r* 88.8%

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

    2. associate-/r/ 88.8%

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

    3. /-rgt-identity 88.8%

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

    4. associate-*r* 91.4%

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

12. Simplified 91.4%

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

    1. associate-*r* 88.8%

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

    2. associate-/r/ 88.8%

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

    3. div-inv 88.8%

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

    4. clear-num 88.8%

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

    5. associate-*l* 88.8%

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

    6. associate-*r/ 91.4%

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

14. Applied egg-rr 91.4%

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

15. Final simplification 91.4%

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

Alternative 2: 88.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{0.5}{\frac{d}{D}}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+233}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot h\right) \cdot \left(M \cdot \frac{0.5 \cdot \frac{D}{d}}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ 0.5 (/ d D)))))
   (if (<= (/ h l) -5e+233)
     (*
      w0
      (sqrt
       (- 1.0 (* (* (* (* M (/ 0.5 d)) D) h) (* M (/ (* 0.5 (/ D d)) l))))))
     (if (<= (/ h l) -1e-323)
       (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_0)))))
       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) <= -5e+233) {
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))));
	} else if ((h / l) <= -1e-323) {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (0.5d0 / (d_1 / d))
    if ((h / l) <= (-5d+233)) then
        tmp = w0 * sqrt((1.0d0 - ((((m * (0.5d0 / d_1)) * d) * h) * (m * ((0.5d0 * (d / d_1)) / l)))))
    else if ((h / l) <= (-1d-323)) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * 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 = M * (0.5 / (d / D));
	double tmp;
	if ((h / l) <= -5e+233) {
		tmp = w0 * Math.sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))));
	} else if ((h / l) <= -1e-323) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	} 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) <= -5e+233:
		tmp = w0 * math.sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))))
	elif (h / l) <= -1e-323:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(0.5 / Float64(d / D)))
	tmp = 0.0
	if (Float64(h / l) <= -5e+233)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(0.5 / d)) * D) * h) * Float64(M * Float64(Float64(0.5 * Float64(D / d)) / l))))));
	elseif (Float64(h / l) <= -1e-323)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	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) <= -5e+233)
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))));
	elseif ((h / l) <= -1e-323)
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -5e+233], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] * N[(M * N[(N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-323], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -5.00000000000000009e233

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

      1. clear-num 68.3%

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

      2. un-div-inv 71.2%

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

      3. *-commutative 71.2%

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

      4. associate-*l/ 71.2%

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

      5. associate-*r/ 71.1%

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

      6. div-inv 71.1%

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

      7. metadata-eval 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*l/ 71.2%

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

      3. associate-/l* 71.2%

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

      4. associate-*r/ 71.2%

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

   8. Simplified 71.2%

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

      1. unpow2 71.2%

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

      2. div-inv 71.3%

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

      3. times-frac 94.4%

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

      4. *-commutative 94.4%

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

      5. associate-*l* 89.0%

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

      6. *-commutative 89.0%

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

      7. associate-*l* 89.0%

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

   10. Applied egg-rr 89.0%

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

       1. associate-*r* 89.0%

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

       2. associate-/r/ 88.9%

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

       3. /-rgt-identity 88.9%

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

       4. associate-*r* 94.3%

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

   12. Simplified 94.3%

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

       1. associate-*r* 88.9%

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

       2. associate-/r/ 89.0%

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

       3. associate-/l* 89.0%

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

       4. div-inv 89.0%

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

       5. clear-num 89.1%

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

   14. Applied egg-rr 89.1%

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

   if -5.00000000000000009e233 < (/.f64 h l) < -9.88131e-324

   1. Initial program 84.9%

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

   3. Simplified 84.8%

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

      1. associate-*r/ 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-*l/ 84.0%

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

      4. associate-*r/ 83.2%

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

      5. div-inv 83.2%

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

      6. metadata-eval 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-/l* 84.0%

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

      2. *-commutative 84.0%

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

      3. unpow2 84.0%

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

      4. associate-*r* 84.8%

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

      5. associate-*l* 84.8%

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

      6. clear-num 84.8%

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

      7. un-div-inv 84.8%

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

      8. associate-*l* 84.8%

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

      9. clear-num 84.8%

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

      10. un-div-inv 84.8%

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

   8. Applied egg-rr 84.8%

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

   if -9.88131e-324 < (/.f64 h l)

   1. Initial program 87.0%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in D around 0 94.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+233}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot h\right) \cdot \left(M \cdot \frac{0.5 \cdot \frac{D}{d}}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-323}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{0.5}{\frac{d}{D}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-240}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot h\right) \cdot \left(M \cdot \frac{0.5 \cdot \frac{D}{d}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.25e-240)
   w0
   (*
    w0
    (sqrt
     (- 1.0 (* (* (* (* M (/ 0.5 d)) D) h) (* M (/ (* 0.5 (/ D d)) l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.25e-240) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / 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 <= 2.25d-240) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((((m * (0.5d0 / d_1)) * d) * h) * (m * ((0.5d0 * (d / d_1)) / 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 <= 2.25e-240) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.25e-240:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.25e-240)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(0.5 / d)) * D) * h) * Float64(M * Float64(Float64(0.5 * Float64(D / d)) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.25e-240)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 / d)) * D) * h) * (M * ((0.5 * (D / d)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.25e-240], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] * N[(M * N[(N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2500000000000001e-240

   1. Initial program 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in D around 0 68.3%

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

   if 2.2500000000000001e-240 < M

   1. Initial program 79.2%

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

   3. Simplified 78.3%

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

      1. clear-num 77.4%

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

      2. un-div-inv 78.3%

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

      3. *-commutative 78.3%

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

      4. associate-*l/ 79.2%

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

      5. associate-*r/ 78.3%

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

      6. div-inv 78.3%

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

      7. metadata-eval 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*l/ 79.2%

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

      3. associate-/l* 78.3%

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

      4. associate-*r/ 78.3%

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

   8. Simplified 78.3%

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

      1. unpow2 78.3%

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

      2. div-inv 78.3%

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

      3. times-frac 90.2%

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

      4. *-commutative 90.2%

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

      5. associate-*l* 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 88.6%

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

   10. Applied egg-rr 88.6%

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

       1. associate-*r* 87.7%

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

       2. associate-/r/ 87.7%

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

       3. /-rgt-identity 87.7%

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

       4. associate-*r* 90.2%

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

   12. Simplified 90.2%

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

       1. associate-*r* 87.7%

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

       2. associate-/r/ 87.7%

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

       3. associate-/l* 85.9%

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

       4. div-inv 85.9%

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

       5. clear-num 85.9%

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

   14. Applied egg-rr 85.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-240}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right) \cdot h\right) \cdot \left(M \cdot \frac{0.5 \cdot \frac{D}{d}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (((M * ((0.5 / d) * D)) / l) * (h * (M * (0.5 * (D / d)))))));
}

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 * ((0.5d0 / d_1) * d)) / l) * (h * (m * (0.5d0 * (d / d_1)))))))
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 - (((M * ((0.5 / d) * D)) / l) * (h * (M * (0.5 * (D / d)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 83.2%

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

   1. clear-num 82.8%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-*l/ 83.6%

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

   5. associate-*r/ 83.2%

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

   6. div-inv 83.2%

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

   7. metadata-eval 83.2%

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

6. Applied egg-rr 83.2%

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

   1. *-commutative 83.2%

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

   2. associate-*l/ 83.6%

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

   3. associate-/l* 83.2%

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

   4. associate-*r/ 83.2%

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

8. Simplified 83.2%

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

   1. unpow2 83.2%

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

   2. div-inv 83.2%

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

   3. times-frac 91.4%

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

   4. *-commutative 91.4%

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

   5. associate-*l* 88.8%

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

   6. *-commutative 88.8%

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

   7. associate-*l* 89.2%

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

10. Applied egg-rr 89.2%

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

    1. associate-/r/ 89.2%

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

    2. /-rgt-identity 89.2%

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

    3. associate-/r/ 89.2%

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

    4. div-inv 89.2%

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

    5. clear-num 89.2%

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

12. Applied egg-rr 89.2%

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

13. Final simplification 89.2%

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

Alternative 5: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot \frac{0.5}{d}\right) \cdot D\\ w0 \cdot \sqrt{1 - \frac{t\_0}{\ell} \cdot \left(t\_0 \cdot h\right)} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (* M (/ 0.5 d)) D)))
   (* w0 (sqrt (- 1.0 (* (/ t_0 l) (* t_0 h)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * (0.5 / d)) * D;
	return w0 * sqrt((1.0 - ((t_0 / l) * (t_0 * h))));
}

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
    t_0 = (m * (0.5d0 / d_1)) * d
    code = w0 * sqrt((1.0d0 - ((t_0 / l) * (t_0 * h))))
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;
	return w0 * Math.sqrt((1.0 - ((t_0 / l) * (t_0 * h))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * (0.5 / d)) * D
	return w0 * math.sqrt((1.0 - ((t_0 / l) * (t_0 * h))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * Float64(0.5 / d)) * D)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(t_0 * h)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = (M * (0.5 / d)) * D;
	tmp = w0 * sqrt((1.0 - ((t_0 / l) * (t_0 * h))));
end

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 83.2%

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

   1. clear-num 82.8%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-*l/ 83.6%

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

   5. associate-*r/ 83.2%

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

   6. div-inv 83.2%

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

   7. metadata-eval 83.2%

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

6. Applied egg-rr 83.2%

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

   1. *-commutative 83.2%

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

   2. associate-*l/ 83.6%

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

   3. associate-/l* 83.2%

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

   4. associate-*r/ 83.2%

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

8. Simplified 83.2%

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

   1. unpow2 83.2%

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

   2. div-inv 83.2%

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

   3. times-frac 91.4%

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

   4. *-commutative 91.4%

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

   5. associate-*l* 88.8%

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

   6. *-commutative 88.8%

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

   7. associate-*l* 89.2%

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

10. Applied egg-rr 89.2%

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

    1. associate-*r* 88.8%

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

    2. associate-/r/ 88.8%

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

    3. /-rgt-identity 88.8%

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

    4. associate-*r* 91.4%

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

12. Simplified 91.4%

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

13. Final simplification 91.4%

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

Alternative 6: 68.5% 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 83.6%

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

3. Simplified 83.2%

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

5. Taylor expanded in D around 0 67.1%

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

6. Final simplification 67.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))))))