Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.9% → 89.5%

Time: 19.1s

Alternatives: 5

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: 81.9% 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: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(M \cdot 0.5\right) \cdot \frac{D}{d}\\ t_1 := \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\\ \mathbf{if}\;h \leq 5 \cdot 10^{-266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{h \cdot t\_0}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1 \cdot \frac{t\_1}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* M 0.5) (/ D d))) (t_1 (* (* (* M D) (/ 0.5 d)) (sqrt h))))
   (if (<= h 5e-266)
     (* w0 (sqrt (- 1.0 (* t_0 (/ (* h t_0) l)))))
     (* w0 (sqrt (- 1.0 (* t_1 (/ t_1 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 t_0 = (M * 0.5) * (D / d);
	double t_1 = ((M * D) * (0.5 / d)) * sqrt(h);
	double tmp;
	if (h <= 5e-266) {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h * t_0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * 0.5d0) * (d / d_1)
    t_1 = ((m * d) * (0.5d0 / d_1)) * sqrt(h)
    if (h <= 5d-266) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h * t_0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_1 * (t_1 / 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 t_0 = (M * 0.5) * (D / d);
	double t_1 = ((M * D) * (0.5 / d)) * Math.sqrt(h);
	double tmp;
	if (h <= 5e-266) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h * t_0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_1 * (t_1 / 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):
	t_0 = (M * 0.5) * (D / d)
	t_1 = ((M * D) * (0.5 / d)) * math.sqrt(h)
	tmp = 0
	if h <= 5e-266:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h * t_0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_1 * (t_1 / 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)
	t_0 = Float64(Float64(M * 0.5) * Float64(D / d))
	t_1 = Float64(Float64(Float64(M * D) * Float64(0.5 / d)) * sqrt(h))
	tmp = 0.0
	if (h <= 5e-266)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h * t_0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_1 * Float64(t_1 / 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)
	t_0 = (M * 0.5) * (D / d);
	t_1 = ((M * D) * (0.5 / d)) * sqrt(h);
	tmp = 0.0;
	if (h <= 5e-266)
		tmp = w0 * sqrt((1.0 - (t_0 * ((h * t_0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 / 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_] := Block[{t$95$0 = N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 5e-266], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 4.99999999999999992e-266

   1. Initial program 79.3%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 86.1%

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

      2. *-commutative 86.1%

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

      3. associate-*l/ 84.5%

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

      4. associate-*r/ 86.1%

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

      5. div-inv 86.1%

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

      6. metadata-eval 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-/l* 80.9%

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

      2. *-commutative 80.9%

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

      3. unpow2 80.9%

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

      4. associate-*r* 83.4%

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

   8. Applied egg-rr 83.4%

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

      1. associate-*l/ 90.2%

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

   10. Applied egg-rr 90.2%

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

   if 4.99999999999999992e-266 < h

   1. Initial program 80.4%

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

   3. Simplified 79.7%

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

      1. associate-/r* 79.7%

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

      2. associate-*r/ 80.4%

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

      3. *-commutative 80.4%

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

      4. associate-*r/ 79.6%

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

      5. unpow2 79.6%

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

      6. unpow2 79.6%

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

      7. associate-*r/ 85.4%

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

      8. clear-num 85.4%

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

      9. associate-*r/ 86.2%

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

      10. times-frac 85.4%

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

      11. div-inv 85.4%

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

      12. metadata-eval 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r/ 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*r/ 86.1%

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

      4. *-commutative 86.1%

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

      5. associate-/l* 85.4%

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

      6. associate-*r/ 85.4%

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

   8. Simplified 85.4%

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

      1. associate-*l/ 85.4%

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

      2. *-un-lft-identity 85.4%

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

   10. Applied egg-rr 85.4%

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

       1. add-sqr-sqrt 85.4%

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

       2. *-un-lft-identity 85.4%

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

       3. times-frac 85.4%

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

       4. *-commutative 85.4%

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

       5. sqrt-prod 85.4%

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

       6. sqrt-pow1 71.8%

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

       7. metadata-eval 71.8%

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

       8. pow1 71.8%

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

       9. associate-*r* 71.1%

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

   12. Applied egg-rr 90.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := \left(M \cdot 0.5\right) \cdot \frac{D}{d}\\ \mathbf{if}\;t\_0 \leq 10^{+129}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1 \cdot \frac{h \cdot t\_1}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l))))
        (t_1 (* (* M 0.5) (/ D d))))
   (if (<= t_0 1e+129)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (* t_1 (/ (* h t_1) 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 t_0 = 1.0 - (pow(((M * D) / (d * 2.0)), 2.0) * (h / l));
	double t_1 = (M * 0.5) * (D / d);
	double tmp;
	if (t_0 <= 1e+129) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (t_1 * ((h * t_1) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))
    t_1 = (m * 0.5d0) * (d / d_1)
    if (t_0 <= 1d+129) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (t_1 * ((h * t_1) / 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 t_0 = 1.0 - (Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l));
	double t_1 = (M * 0.5) * (D / d);
	double tmp;
	if (t_0 <= 1e+129) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_1 * ((h * t_1) / 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):
	t_0 = 1.0 - (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))
	t_1 = (M * 0.5) * (D / d)
	tmp = 0
	if t_0 <= 1e+129:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (t_1 * ((h * t_1) / 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)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)))
	t_1 = Float64(Float64(M * 0.5) * Float64(D / d))
	tmp = 0.0
	if (t_0 <= 1e+129)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_1 * Float64(Float64(h * t_1) / 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)
	t_0 = 1.0 - ((((M * D) / (d * 2.0)) ^ 2.0) * (h / l));
	t_1 = (M * 0.5) * (D / d);
	tmp = 0.0;
	if (t_0 <= 1e+129)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (t_1 * ((h * t_1) / 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_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+129], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$1 * N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1e129

   1. Initial program 99.9%

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

   if 1e129 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 38.9%

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

   3. Simplified 42.3%

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

      1. associate-*r/ 61.4%

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

      2. *-commutative 61.4%

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

      3. associate-*l/ 57.9%

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

      4. associate-*r/ 61.4%

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

      5. div-inv 61.4%

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

      6. metadata-eval 61.4%

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

   6. Applied egg-rr 61.4%

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

      1. associate-/l* 42.3%

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

      2. *-commutative 42.3%

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

      3. unpow2 42.3%

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

      4. associate-*r* 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. associate-*l/ 69.3%

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

   10. Applied egg-rr 69.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+129}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 1.8× speedup

Math

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

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 (/ (* D (* M (/ 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 * ((D * (M * (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 / d_1)) * (h * ((d * (m * (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 * ((D * (M * (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 * ((D * (M * (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(Float64(M * 0.5) * Float64(D / d)) * Float64(h * Float64(Float64(D * Float64(M * 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 * ((D * (M * (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[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Simplified 80.2%

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

   1. associate-*r/ 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*l/ 85.4%

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

   4. associate-*r/ 85.7%

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

   5. div-inv 85.7%

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

   6. metadata-eval 85.7%

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

6. Applied egg-rr 85.7%

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

   1. associate-/l* 80.2%

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

   2. *-commutative 80.2%

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

   3. unpow2 80.2%

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

   4. associate-*r* 82.1%

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

8. Applied egg-rr 82.1%

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

   1. associate-*l/ 88.7%

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

10. Applied egg-rr 88.7%

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

    1. associate-/l* 88.3%

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

    2. *-commutative 88.3%

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

    3. associate-*l/ 87.2%

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

    4. associate-*r/ 87.6%

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

    5. associate-*r/ 87.6%

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

12. Simplified 87.6%

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

13. Final simplification 87.6%

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

Alternative 4: 89.3% accurate, 1.8× speedup

Math

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

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
 (let* ((t_0 (* (* M 0.5) (/ D d))))
   (* w0 (sqrt (- 1.0 (* t_0 (/ (* h t_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 t_0 = (M * 0.5) * (D / d);
	return w0 * sqrt((1.0 - (t_0 * ((h * t_0) / 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
    real(8) :: t_0
    t_0 = (m * 0.5d0) * (d / d_1)
    code = w0 * sqrt((1.0d0 - (t_0 * ((h * t_0) / 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) {
	double t_0 = (M * 0.5) * (D / d);
	return w0 * Math.sqrt((1.0 - (t_0 * ((h * t_0) / l))));
}

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 * 0.5) * (D / d)
	return w0 * math.sqrt((1.0 - (t_0 * ((h * t_0) / l))))

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(Float64(M * 0.5) * Float64(D / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h * t_0) / 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)
	t_0 = (M * 0.5) * (D / d);
	tmp = w0 * sqrt((1.0 - (t_0 * ((h * t_0) / 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_] := Block[{t$95$0 = N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.9%

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

3. Simplified 80.2%

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

   1. associate-*r/ 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*l/ 85.4%

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

   4. associate-*r/ 85.7%

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

   5. div-inv 85.7%

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

   6. metadata-eval 85.7%

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

6. Applied egg-rr 85.7%

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

   1. associate-/l* 80.2%

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

   2. *-commutative 80.2%

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

   3. unpow2 80.2%

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

   4. associate-*r* 82.1%

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

8. Applied egg-rr 82.1%

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

   1. associate-*l/ 88.7%

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

10. Applied egg-rr 88.7%

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

11. Final simplification 88.7%

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

Alternative 5: 67.8% 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 79.9%

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

3. Simplified 80.2%

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

5. Taylor expanded in D around 0 67.6%

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

6. Final simplification 67.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024084 
(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))))))