Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.9%

Time: 16.8s

Alternatives: 8

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.9% accurate, 1.7× speedup

Math

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

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 (* D (/ 0.5 d))) (* (* M (* 0.5 (/ D d))) 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) {
	return w0 * sqrt((1.0 + (((M * (D * (0.5 / d))) * ((M * (0.5 * (D / d))) * h)) * (-1.0 / 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
    code = w0 * sqrt((1.0d0 + (((m * (d * (0.5d0 / d_1))) * ((m * (0.5d0 * (d / d_1))) * h)) * ((-1.0d0) / 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 * (D * (0.5 / d))) * ((M * (0.5 * (D / d))) * h)) * (-1.0 / 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 * (D * (0.5 / d))) * ((M * (0.5 * (D / d))) * h)) * (-1.0 / 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 * Float64(D * Float64(0.5 / d))) * Float64(Float64(M * Float64(0.5 * Float64(D / d))) * h)) * Float64(-1.0 / 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)
	tmp = w0 * sqrt((1.0 + (((M * (D * (0.5 / d))) * ((M * (0.5 * (D / d))) * h)) * (-1.0 / 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_] := N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

3. Simplified 81.3%

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

   1. unpow2 81.3%

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

   2. *-commutative 81.3%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

   5. associate-*r* 79.9%

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

   6. *-commutative 79.9%

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

   7. associate-*l/ 79.1%

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

   8. associate-*r/ 79.9%

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

   9. times-frac 79.1%

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

   10. associate-/l* 79.9%

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

   11. div-inv 79.9%

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

   12. associate-/r* 79.9%

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

   13. metadata-eval 79.9%

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

   14. div-inv 79.9%

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

   15. metadata-eval 79.9%

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

6. Applied egg-rr 79.9%

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

   1. associate-*r/ 85.0%

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

   2. associate-*l* 85.7%

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

   3. associate-*r* 85.0%

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

8. Applied egg-rr 85.0%

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

   1. div-inv 85.0%

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

   2. associate-*l* 89.2%

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

   3. associate-*l* 90.3%

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

   4. associate-*l* 90.3%

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

10. Applied egg-rr 90.3%

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

11. Final simplification 90.3%

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

Alternative 2: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(M \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) -2e-278)
   (*
    w0
    (sqrt (- 1.0 (* (/ h l) (* (/ D d) (* (* M (* D (/ 0.5 d))) (* M 0.5)))))))
   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) {
	double tmp;
	if ((h / l) <= -2e-278) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / d) * ((M * (D * (0.5 / d))) * (M * 0.5))))));
	} else {
		tmp = w0;
	}
	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 ((h / l) <= (-2d-278)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / d_1) * ((m * (d * (0.5d0 / d_1))) * (m * 0.5d0))))))
    else
        tmp = w0
    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 ((h / l) <= -2e-278) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((D / d) * ((M * (D * (0.5 / d))) * (M * 0.5))))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= -2e-278:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((D / d) * ((M * (D * (0.5 / d))) * (M * 0.5))))))
	else:
		tmp = w0
	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(h / l) <= -2e-278)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(D / d) * Float64(Float64(M * Float64(D * Float64(0.5 / d))) * Float64(M * 0.5)))))));
	else
		tmp = w0;
	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 ((h / l) <= -2e-278)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / d) * ((M * (D * (0.5 / d))) * (M * 0.5))))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], -2e-278], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999999999988e-278

   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 81.8%

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

      1. unpow2 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*l/ 80.5%

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

      4. associate-*r/ 81.2%

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

      5. associate-*r* 80.5%

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

      6. *-commutative 80.5%

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

      7. associate-*l/ 79.2%

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

      8. associate-*r/ 80.5%

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

      9. times-frac 79.2%

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

      10. associate-/l* 80.5%

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

      11. div-inv 80.5%

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

      12. associate-/r* 80.5%

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

      13. metadata-eval 80.5%

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

      14. div-inv 80.5%

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

      15. metadata-eval 80.5%

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

   6. Applied egg-rr 80.5%

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

   if -1.99999999999999988e-278 < (/.f64 h l)

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in D around 0 92.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(M \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{\frac{h}{\ell}}{{d}^{2}} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

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 (<= M 1.05e+58)
   w0
   (* w0 (* (* (* M D) (* M D)) (* (/ (/ h l) (pow d 2.0)) -0.125)))))

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 <= 1.05e+58) {
		tmp = w0;
	} else {
		tmp = w0 * (((M * D) * (M * D)) * (((h / l) / pow(d, 2.0)) * -0.125));
	}
	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 <= 1.05d+58) then
        tmp = w0
    else
        tmp = w0 * (((m * d) * (m * d)) * (((h / l) / (d_1 ** 2.0d0)) * (-0.125d0)))
    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 <= 1.05e+58) {
		tmp = w0;
	} else {
		tmp = w0 * (((M * D) * (M * D)) * (((h / l) / Math.pow(d, 2.0)) * -0.125));
	}
	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 <= 1.05e+58:
		tmp = w0
	else:
		tmp = w0 * (((M * D) * (M * D)) * (((h / l) / math.pow(d, 2.0)) * -0.125))
	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 (M <= 1.05e+58)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(Float64(Float64(h / l) / (d ^ 2.0)) * -0.125)));
	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 <= 1.05e+58)
		tmp = w0;
	else
		tmp = w0 * (((M * D) * (M * D)) * (((h / l) / (d ^ 2.0)) * -0.125));
	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[M, 1.05e+58], w0, N[(w0 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.05000000000000006e58

   1. Initial program 81.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in D around 0 76.0%

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

   if 1.05000000000000006e58 < M

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in D around 0 55.0%

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

      1. +-commutative 55.0%

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

      2. *-commutative 55.0%

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

      3. fma-define 55.0%

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

      4. associate-*r* 57.2%

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

      5. unpow2 57.2%

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

      6. unpow2 57.2%

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

      7. swap-sqr 68.1%

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

      8. unpow2 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in D around inf 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-*r* 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

      5. unpow2 40.2%

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

      6. swap-sqr 43.0%

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

      7. unpow2 43.0%

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

      8. *-commutative 43.0%

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

      9. associate-*r/ 38.8%

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

      10. associate-*l* 38.8%

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

      11. *-commutative 38.8%

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

      12. associate-/r* 36.5%

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

   10. Simplified 36.5%

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

       1. unpow2 36.5%

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

       2. *-commutative 36.5%

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

       3. *-commutative 36.5%

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

   12. Applied egg-rr 36.5%

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

Alternative 4: 85.4% accurate, 1.8× speedup

Math

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

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 (* (* (/ 0.5 d) (* M D)) (* (/ D d) (* M 0.5)))) 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 - ((h * (((0.5 / d) * (M * D)) * ((D / d) * (M * 0.5)))) / 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 - ((h * (((0.5d0 / d_1) * (m * d)) * ((d / d_1) * (m * 0.5d0)))) / 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 - ((h * (((0.5 / d) * (M * D)) * ((D / d) * (M * 0.5)))) / 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 - ((h * (((0.5 / d) * (M * D)) * ((D / d) * (M * 0.5)))) / 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(h * Float64(Float64(Float64(0.5 / d) * Float64(M * D)) * Float64(Float64(D / d) * Float64(M * 0.5)))) / 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 - ((h * (((0.5 / d) * (M * D)) * ((D / d) * (M * 0.5)))) / 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[(h * N[(N[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

3. Simplified 81.3%

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

   1. unpow2 81.3%

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

   2. *-commutative 81.3%

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

   3. associate-*l/ 80.6%

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

   4. associate-*r/ 80.6%

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

   5. associate-*r* 79.9%

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

   6. *-commutative 79.9%

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

   7. associate-*l/ 79.1%

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

   8. associate-*r/ 79.9%

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

   9. times-frac 79.1%

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

   10. associate-/l* 79.9%

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

   11. div-inv 79.9%

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

   12. associate-/r* 79.9%

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

   13. metadata-eval 79.9%

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

   14. div-inv 79.9%

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

   15. metadata-eval 79.9%

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

6. Applied egg-rr 79.9%

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

   1. associate-*r/ 85.0%

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

   2. associate-*l* 85.7%

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

   3. associate-*r* 85.0%

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

8. Applied egg-rr 85.0%

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

9. Final simplification 85.0%

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

Alternative 5: 68.2% accurate, 18.0× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{+90}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\ell}{w0 \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= M 1.3e+90) w0 (/ 1.0 (/ l (* 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 (M <= 1.3e+90) {
		tmp = w0;
	} else {
		tmp = 1.0 / (l / (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 (m <= 1.3d+90) then
        tmp = w0
    else
        tmp = 1.0d0 / (l / (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 (M <= 1.3e+90) {
		tmp = w0;
	} else {
		tmp = 1.0 / (l / (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 M <= 1.3e+90:
		tmp = w0
	else:
		tmp = 1.0 / (l / (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 (M <= 1.3e+90)
		tmp = w0;
	else
		tmp = Float64(1.0 / Float64(l / 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 (M <= 1.3e+90)
		tmp = w0;
	else
		tmp = 1.0 / (l / (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[M, 1.3e+90], w0, N[(1.0 / N[(l / N[(w0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.2999999999999999e90

   1. Initial program 81.4%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in D around 0 75.2%

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

   if 1.2999999999999999e90 < M

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in D around 0 54.4%

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

      1. +-commutative 54.4%

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

      2. *-commutative 54.4%

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

      3. fma-define 54.4%

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

      4. associate-*r* 54.4%

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

      5. unpow2 54.4%

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

      6. unpow2 54.4%

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

      7. swap-sqr 67.9%

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

      8. unpow2 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in l around 0 47.7%

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

      1. fma-define 47.7%

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

      2. associate-*r* 54.4%

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

      3. *-commutative 54.4%

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

      4. unpow2 54.4%

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

      5. unpow2 54.4%

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

      6. swap-sqr 68.1%

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

      7. unpow2 68.1%

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

      8. *-commutative 68.1%

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

      9. *-commutative 68.1%

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

      10. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in D around 0 34.4%

       \[\leadsto \frac{\color{blue}{\ell \cdot w0}}{\ell} \]
   12. Step-by-step derivation

       1. clear-num 34.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{\ell}{\ell \cdot w0}}} \]

       2. inv-pow 34.3%

          \[\leadsto \color{blue}{{\left(\frac{\ell}{\ell \cdot w0}\right)}^{-1}} \]

   13. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{\ell \cdot w0}\right)}^{-1}} \]
   14. Step-by-step derivation

       1. unpow-1 34.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{\ell}{\ell \cdot w0}}} \]

       2. *-commutative 34.3%

          \[\leadsto \frac{1}{\frac{\ell}{\color{blue}{w0 \cdot \ell}}} \]

   15. Simplified 34.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{\ell}{w0 \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{+64}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{w0 \cdot \ell}{\ell}\\ \end{array} \end{array} \]

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 (<= M 7.8e+64) w0 (/ (* w0 l) 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 <= 7.8e+64) {
		tmp = w0;
	} else {
		tmp = (w0 * l) / 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 <= 7.8d+64) then
        tmp = w0
    else
        tmp = (w0 * l) / 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 <= 7.8e+64) {
		tmp = w0;
	} else {
		tmp = (w0 * l) / 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 <= 7.8e+64:
		tmp = w0
	else:
		tmp = (w0 * l) / 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 (M <= 7.8e+64)
		tmp = w0;
	else
		tmp = Float64(Float64(w0 * l) / 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 <= 7.8e+64)
		tmp = w0;
	else
		tmp = (w0 * l) / 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[M, 7.8e+64], w0, N[(N[(w0 * l), $MachinePrecision] / l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.7999999999999996e64

   1. Initial program 81.1%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in D around 0 75.3%

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

   if 7.7999999999999996e64 < M

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in D around 0 52.8%

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

      1. +-commutative 52.8%

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

      2. *-commutative 52.8%

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

      3. fma-define 52.8%

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

      4. associate-*r* 55.3%

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

      5. unpow2 55.3%

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

      6. unpow2 55.3%

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

      7. swap-sqr 67.8%

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

      8. unpow2 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in l around 0 46.6%

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

      1. fma-define 46.6%

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

      2. associate-*r* 52.8%

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

      3. *-commutative 52.8%

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

      4. unpow2 52.8%

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

      5. unpow2 52.8%

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

      6. swap-sqr 65.4%

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

      7. unpow2 65.4%

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

      8. *-commutative 65.4%

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

      9. *-commutative 65.4%

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

      10. *-commutative 65.4%

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

   10. Simplified 65.4%

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

   11. Taylor expanded in D around 0 34.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{+64}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{w0 \cdot \ell}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{w0}{\ell}\\ \end{array} \end{array} \]

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 (<= M 3.6e+25) w0 (* l (/ 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 (M <= 3.6e+25) {
		tmp = w0;
	} else {
		tmp = l * (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 (m <= 3.6d+25) then
        tmp = w0
    else
        tmp = l * (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 (M <= 3.6e+25) {
		tmp = w0;
	} else {
		tmp = l * (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 M <= 3.6e+25:
		tmp = w0
	else:
		tmp = l * (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 (M <= 3.6e+25)
		tmp = w0;
	else
		tmp = Float64(l * 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 (M <= 3.6e+25)
		tmp = w0;
	else
		tmp = l * (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[M, 3.6e+25], w0, N[(l * N[(w0 / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.60000000000000015e25

   1. Initial program 81.6%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in D around 0 75.7%

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

   if 3.60000000000000015e25 < M

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in D around 0 56.6%

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

      1. +-commutative 56.6%

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

      2. *-commutative 56.6%

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

      3. fma-define 56.6%

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

      4. associate-*r* 58.6%

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

      5. unpow2 58.6%

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

      6. unpow2 58.6%

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

      7. swap-sqr 68.6%

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

      8. unpow2 68.6%

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

   7. Simplified 68.6%

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

   8. Taylor expanded in l around 0 45.8%

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

      1. fma-define 45.8%

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

      2. associate-*r* 50.8%

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

      3. *-commutative 50.8%

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

      4. unpow2 50.8%

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

      5. unpow2 50.8%

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

      6. swap-sqr 60.9%

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

      7. unpow2 60.9%

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

      8. *-commutative 60.9%

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

      9. *-commutative 60.9%

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

      10. *-commutative 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in D around 0 35.5%

       \[\leadsto \frac{\color{blue}{\ell \cdot w0}}{\ell} \]
   12. Step-by-step derivation

       1. associate-/l* 43.1%

          \[\leadsto \color{blue}{\ell \cdot \frac{w0}{\ell}} \]

   13. Applied egg-rr 43.1%

       \[\leadsto \color{blue}{\ell \cdot \frac{w0}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.0% 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 80.6%

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

3. Simplified 81.3%

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

5. Taylor expanded in D around 0 69.0%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))))))