Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.3%

Time: 18.3s

Alternatives: 10

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: 81.0% 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: 89.3% accurate, 1.8× speedup

Math

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

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (/ (* h (* (* M 0.5) (/ D d))) l) (* M (/ 0.5 (/ d D))))))))

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 * ((m * 0.5d0) * (d / d_1))) / l) * (m * (0.5d0 / (d_1 / d))))))
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (((h * ((M * 0.5) * (D / d))) / l) * (M * (0.5 / (d / D))))));
}

Python

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

Julia

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

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (((h * ((M * 0.5) * (D / d))) / l) * (M * (0.5 / (d / D))))));
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Simplified 81.8%

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

   1. associate-*r/ 86.7%

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

   2. add-sqr-sqrt 86.7%

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

   3. pow2 86.7%

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

   4. unpow2 86.7%

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

   5. sqrt-prod 51.6%

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

   6. add-sqr-sqrt 86.7%

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

   7. *-un-lft-identity 86.7%

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

   8. times-frac 86.7%

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

   9. metadata-eval 86.7%

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

6. Applied egg-rr 86.7%

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

   1. associate-*r/ 81.8%

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

   2. *-commutative 81.8%

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

   3. unpow2 81.8%

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

   4. associate-*r* 84.1%

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

   5. clear-num 84.1%

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

   6. un-div-inv 84.1%

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

   7. clear-num 84.1%

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

   8. un-div-inv 84.1%

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

8. Applied egg-rr 84.1%

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

   1. associate-*l/ 91.1%

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

   2. associate-*r/ 91.1%

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

   3. div-inv 91.1%

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

   4. clear-num 91.1%

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

10. Applied egg-rr 91.1%

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

11. Final simplification 91.1%

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

Alternative 2: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(w0 \cdot h\right)\right) \cdot \frac{1}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 5e-19)
   w0
   (+ w0 (* -0.125 (* (* (pow (* M (/ D d)) 2.0) (* w0 h)) (/ 1.0 l))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 5e-19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((pow((M * (D / d)), 2.0) * (w0 * h)) * (1.0 / l)));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 * d) <= 5d-19) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m * (d / d_1)) ** 2.0d0) * (w0 * h)) * (1.0d0 / l)))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 5e-19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((Math.pow((M * (D / d)), 2.0) * (w0 * h)) * (1.0 / l)));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M * D) <= 5e-19:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((math.pow((M * (D / d)), 2.0) * (w0 * h)) * (1.0 / l)))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 5e-19)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64((Float64(M * Float64(D / d)) ^ 2.0) * Float64(w0 * h)) * Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M * D) <= 5e-19)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M * (D / d)) ^ 2.0) * (w0 * h)) * (1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 5e-19], w0, N[(w0 + N[(-0.125 * N[(N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000004e-19

   1. Initial program 84.0%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in M around 0 74.8%

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

   if 5.0000000000000004e-19 < (*.f64 M D)

   1. Initial program 73.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in M around 0 41.4%

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

      1. times-frac 43.1%

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

   7. Simplified 43.1%

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

      1. associate-*r/ 43.2%

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

      2. unpow2 43.2%

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

      3. unpow2 43.2%

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

      4. frac-times 62.1%

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

      5. pow2 62.1%

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

   9. Applied egg-rr 62.1%

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

       1. div-inv 62.1%

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

       2. associate-*r* 62.1%

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

       3. pow-prod-down 66.4%

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

   11. Applied egg-rr 66.4%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{1}{\ell}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(w0 \cdot h\right)\right) \cdot \frac{1}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 1.8× speedup

Math

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

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (* M (/ 0.5 (/ d D))) (* h (* M (/ (* D (/ 0.5 d)) l))))))))

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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))) * (h * (m * ((d * (0.5d0 / d_1)) / l))))))
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
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))) * (h * (M * ((D * (0.5 / d)) / l))))));
}

Python

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

Julia

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

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((M * (0.5 / (d / D))) * (h * (M * ((D * (0.5 / d)) / l))))));
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * N[(N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Simplified 81.8%

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

   1. associate-*r/ 86.7%

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

   2. add-sqr-sqrt 86.7%

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

   3. pow2 86.7%

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

   4. unpow2 86.7%

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

   5. sqrt-prod 51.6%

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

   6. add-sqr-sqrt 86.7%

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

   7. *-un-lft-identity 86.7%

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

   8. times-frac 86.7%

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

   9. metadata-eval 86.7%

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

6. Applied egg-rr 86.7%

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

   1. associate-*r/ 81.8%

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

   2. *-commutative 81.8%

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

   3. unpow2 81.8%

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

   4. associate-*r* 84.1%

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

   5. clear-num 84.1%

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

   6. un-div-inv 84.1%

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

   7. clear-num 84.1%

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

   8. un-div-inv 84.1%

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

8. Applied egg-rr 84.1%

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

   1. associate-*l/ 91.1%

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

   2. associate-*r/ 91.1%

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

   3. div-inv 91.1%

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

   4. clear-num 91.1%

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

10. Applied egg-rr 91.1%

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

    1. associate-*l* 91.1%

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

    2. clear-num 91.1%

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

    3. div-inv 91.1%

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

    4. associate-/l* 89.6%

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

    5. associate-*r/ 89.6%

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

12. Applied egg-rr 89.6%

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

    1. associate-/l* 89.6%

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

    2. associate-/l* 87.6%

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

    3. associate-/r/ 87.7%

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

14. Simplified 87.7%

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

15. Final simplification 87.7%

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

Alternative 4: 89.3% accurate, 1.8× speedup

Math

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

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (* M (/ 0.5 (/ d D))) (* h (/ (/ (* M 0.5) (/ d D)) l)))))))

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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))) * (h * (((m * 0.5d0) / (d_1 / d)) / l)))))
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
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))) * (h * (((M * 0.5) / (d / D)) / l)))));
}

Python

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

Julia

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

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((M * (0.5 / (d / D))) * (h * (((M * 0.5) / (d / D)) / l)))));
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Simplified 81.8%

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

   1. associate-*r/ 86.7%

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

   2. add-sqr-sqrt 86.7%

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

   3. pow2 86.7%

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

   4. unpow2 86.7%

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

   5. sqrt-prod 51.6%

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

   6. add-sqr-sqrt 86.7%

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

   7. *-un-lft-identity 86.7%

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

   8. times-frac 86.7%

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

   9. metadata-eval 86.7%

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

6. Applied egg-rr 86.7%

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

   1. associate-*r/ 81.8%

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

   2. *-commutative 81.8%

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

   3. unpow2 81.8%

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

   4. associate-*r* 84.1%

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

   5. clear-num 84.1%

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

   6. un-div-inv 84.1%

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

   7. clear-num 84.1%

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

   8. un-div-inv 84.1%

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

8. Applied egg-rr 84.1%

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

   1. associate-*l/ 91.1%

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

   2. associate-*r/ 91.1%

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

   3. div-inv 91.1%

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

   4. clear-num 91.1%

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

10. Applied egg-rr 91.1%

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

    1. associate-*l* 91.1%

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

    2. clear-num 91.1%

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

    3. div-inv 91.1%

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

    4. associate-/l* 89.6%

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

    5. associate-*r/ 89.6%

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

12. Applied egg-rr 89.6%

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

13. Final simplification 89.6%

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

Alternative 5: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 4.2e+18)
   w0
   (+ w0 (* -0.125 (* (pow (* D (/ M d)) 2.0) (* h (/ w0 l)))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.2e+18) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D * (M / d)), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 <= 4.2d+18) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * (m / d_1)) ** 2.0d0) * (h * (w0 / l))))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.2e+18) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D * (M / d)), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 4.2e+18:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D * (M / d)), 2.0) * (h * (w0 / l))))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.2e+18)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(h * Float64(w0 / l)))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 4.2e+18)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * (M / d)) ^ 2.0) * (h * (w0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4.2e+18], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.2e18

   1. Initial program 85.3%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in M around 0 75.0%

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

   if 4.2e18 < D

   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 68.4%

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

   5. Taylor expanded in M around 0 35.7%

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

      1. times-frac 37.7%

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

   7. Simplified 37.7%

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

      1. unpow2 37.7%

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

      2. unpow2 37.7%

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

      3. frac-times 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in D around 0 35.7%

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

       1. times-frac 37.7%

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

       2. associate-*r/ 37.6%

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

       3. unpow2 37.6%

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

       4. unpow2 37.6%

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

       5. times-frac 57.1%

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

       6. unpow2 57.1%

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

       7. associate-*r* 57.0%

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

       8. unpow2 57.0%

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

       9. unpow2 57.0%

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

       10. swap-sqr 58.3%

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

       11. unpow2 58.3%

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

       12. associate-*l/ 58.2%

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

       13. associate-/l* 54.1%

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

   12. Simplified 54.1%

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

   13. Taylor expanded in D around 0 37.5%

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

       1. *-commutative 37.5%

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

       2. associate-/l* 37.5%

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

       3. unpow2 37.5%

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

       4. unpow2 37.5%

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

       5. unpow2 37.5%

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

       6. times-frac 53.0%

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

       7. swap-sqr 54.3%

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

       8. unpow2 54.3%

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

       9. *-commutative 54.3%

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

       10. associate-*l/ 54.1%

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

       11. associate-*r/ 53.6%

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

   15. Simplified 53.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 6.9 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 6.9e+19)
   w0
   (+ w0 (* -0.125 (* (pow (/ (* M D) d) 2.0) (* h (/ w0 l)))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6.9e+19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow(((M * D) / d), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 <= 6.9d+19) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m * d) / d_1) ** 2.0d0) * (h * (w0 / l))))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6.9e+19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow(((M * D) / d), 2.0) * (h * (w0 / l))));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 6.9e+19:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow(((M * D) / d), 2.0) * (h * (w0 / l))))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 6.9e+19)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) * Float64(h * Float64(w0 / l)))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 6.9e+19)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M * D) / d) ^ 2.0) * (h * (w0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 6.9e+19], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.9e19

   1. Initial program 85.3%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in M around 0 75.0%

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

   if 6.9e19 < D

   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 68.4%

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

   5. Taylor expanded in M around 0 35.7%

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

      1. times-frac 37.7%

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

   7. Simplified 37.7%

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

      1. unpow2 37.7%

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

      2. unpow2 37.7%

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

      3. frac-times 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in D around 0 35.7%

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

       1. times-frac 37.7%

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

       2. associate-*r/ 37.6%

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

       3. unpow2 37.6%

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

       4. unpow2 37.6%

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

       5. times-frac 57.1%

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

       6. unpow2 57.1%

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

       7. associate-*r* 57.0%

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

       8. unpow2 57.0%

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

       9. unpow2 57.0%

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

       10. swap-sqr 58.3%

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

       11. unpow2 58.3%

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

       12. associate-*l/ 58.2%

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

       13. associate-/l* 54.1%

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

   12. Simplified 54.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.9 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 1.2e+19)
   w0
   (+ w0 (* -0.125 (/ (* (* w0 h) (pow (* D (/ M d)) 2.0)) l)))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.2e+19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * pow((D * (M / d)), 2.0)) / l));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 <= 1.2d+19) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((w0 * h) * ((d * (m / d_1)) ** 2.0d0)) / l))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.2e+19) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * Math.pow((D * (M / d)), 2.0)) / l));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1.2e+19:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((w0 * h) * math.pow((D * (M / d)), 2.0)) / l))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1.2e+19)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(w0 * h) * (Float64(D * Float64(M / d)) ^ 2.0)) / l)));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1.2e+19)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((w0 * h) * ((D * (M / d)) ^ 2.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 1.2e+19], w0, N[(w0 + N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.2e19

   1. Initial program 85.3%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in M around 0 75.0%

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

   if 1.2e19 < D

   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 68.4%

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

   5. Taylor expanded in M around 0 35.7%

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

      1. times-frac 37.7%

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

   7. Simplified 37.7%

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

      1. associate-*r/ 37.8%

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

      2. unpow2 37.8%

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

      3. unpow2 37.8%

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

      4. frac-times 59.3%

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

      5. pow2 59.3%

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

   9. Applied egg-rr 59.3%

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

       1. pow1 59.3%

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

       2. associate-*r* 59.3%

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

       3. pow-prod-down 60.0%

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

   11. Applied egg-rr 60.0%

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

       1. unpow1 60.0%

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

       2. *-commutative 60.0%

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

       3. associate-*l/ 59.8%

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

       4. associate-/l* 60.0%

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

   13. Simplified 60.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 6 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 6e+17)
   w0
   (+ w0 (* -0.125 (/ (* (* w0 h) (pow (/ (* M D) d) 2.0)) l)))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6e+17) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * pow(((M * D) / d), 2.0)) / l));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 <= 6d+17) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((w0 * h) * (((m * d) / d_1) ** 2.0d0)) / l))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6e+17) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) * Math.pow(((M * D) / d), 2.0)) / l));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 6e+17:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((w0 * h) * math.pow(((M * D) / d), 2.0)) / l))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 6e+17)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(w0 * h) * (Float64(Float64(M * D) / d) ^ 2.0)) / l)));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 6e+17)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((w0 * h) * (((M * D) / d) ^ 2.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 6e+17], w0, N[(w0 + N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6e17

   1. Initial program 85.3%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in M around 0 75.0%

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

   if 6e17 < D

   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 68.4%

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

   5. Taylor expanded in M around 0 35.7%

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

      1. times-frac 37.7%

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

   7. Simplified 37.7%

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

      1. associate-*r/ 37.8%

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

      2. unpow2 37.8%

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

      3. unpow2 37.8%

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

      4. frac-times 59.3%

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

      5. pow2 59.3%

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

   9. Applied egg-rr 59.3%

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

       1. pow1 59.3%

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

       2. associate-*r* 59.3%

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

       3. pow-prod-down 60.0%

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

   11. Applied egg-rr 60.0%

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

       1. unpow1 60.0%

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

       2. *-commutative 60.0%

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

       3. associate-*l/ 59.8%

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

   13. Simplified 59.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;D \leq 1.4 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= D 1.4e+51) w0 (+ w0 (* -0.125 (* (* h (/ w0 l)) (* t_0 t_0)))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if (D <= 1.4e+51) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * (t_0 * t_0)));
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / d_1)
    if (d <= 1.4d+51) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * (w0 / l)) * (t_0 * t_0)))
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if (D <= 1.4e+51) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * (t_0 * t_0)));
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	t_0 = M * (D / d)
	tmp = 0
	if D <= 1.4e+51:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * (t_0 * t_0)))
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (D <= 1.4e+51)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 / l)) * Float64(t_0 * t_0))));
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M * (D / d);
	tmp = 0.0;
	if (D <= 1.4e+51)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * (w0 / l)) * (t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 1.4e+51], w0, N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.40000000000000002e51

   1. Initial program 85.5%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in M around 0 74.0%

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

   if 1.40000000000000002e51 < D

   1. Initial program 66.2%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in M around 0 35.6%

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

      1. times-frac 37.7%

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

   7. Simplified 37.7%

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

      1. unpow2 37.7%

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

      2. unpow2 37.7%

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

      3. frac-times 60.6%

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

   9. Applied egg-rr 60.6%

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

   10. Taylor expanded in D around 0 35.6%

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

       1. times-frac 37.7%

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

       2. associate-*r/ 37.6%

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

       3. unpow2 37.6%

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

       4. unpow2 37.6%

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

       5. times-frac 58.4%

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

       6. unpow2 58.4%

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

       7. associate-*r* 58.2%

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

       8. unpow2 58.2%

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

       9. unpow2 58.2%

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

       10. swap-sqr 59.7%

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

       11. unpow2 59.7%

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

       12. associate-*l/ 59.5%

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

       13. associate-/l* 55.2%

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

   12. Simplified 55.2%

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

       1. associate-*l/ 55.4%

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

       2. unpow2 55.4%

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

       3. *-commutative 55.4%

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

       4. *-commutative 55.4%

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

   14. Applied egg-rr 55.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.4 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \frac{w0}{\ell}\right) \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ w0 \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	return w0

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 82.1%

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

3. Simplified 81.8%

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

5. Taylor expanded in M around 0 68.2%

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

6. Final simplification 68.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))))