Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 87.8%

Time: 17.5s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{\left(\frac{M\_m}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot h}{\left(\frac{2}{M\_m} \cdot \frac{d}{D}\right) \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (/ (* (* (/ M_m d) (* D 0.5)) h) (* (* (/ 2.0 M_m) (/ d D)) l))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((((M_m / d) * (D * 0.5)) * h) / (((2.0 / M_m) * (d / D)) * l))));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m / d_1) * (d * 0.5d0)) * h) / (((2.0d0 / m_m) * (d_1 / d)) * l))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((((M_m / d) * (D * 0.5)) * h) / (((2.0 / M_m) * (d / D)) * l))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((((M_m / d) * (D * 0.5)) * h) / (((2.0 / M_m) * (d / D)) * l))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(D * 0.5)) * h) / Float64(Float64(Float64(2.0 / M_m) * Float64(d / D)) * l)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M_m / d) * (D * 0.5)) * h) / (((2.0 / M_m) * (d / D)) * l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[(2.0 / M$95$m), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*r/ 81.4%

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

   3. clear-num 81.4%

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

   4. un-div-inv 81.4%

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

   5. *-un-lft-identity 81.4%

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

   6. times-frac 81.4%

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

   7. metadata-eval 81.4%

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

   8. times-frac 82.9%

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

6. Applied egg-rr 82.9%

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

   1. frac-times 90.2%

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

   2. associate-*r/ 90.2%

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

8. Applied egg-rr 90.2%

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

   1. clear-num 90.3%

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

   2. inv-pow 90.3%

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

10. Applied egg-rr 90.3%

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

    1. unpow-1 90.3%

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

12. Simplified 90.3%

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

    1. un-div-inv 90.2%

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

14. Applied egg-rr 90.2%

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

    1. associate-/r/ 88.7%

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

    2. *-commutative 88.7%

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

16. Simplified 88.7%

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

Alternative 2: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D \leq 2 \cdot 10^{-208}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M\_m \cdot 0.5}{d}\right) \cdot \frac{D \cdot \frac{h}{d}}{\frac{2}{M\_m} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 2e-208)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* D (/ (* M_m 0.5) d)) (/ (* D (/ h d)) (* (/ 2.0 M_m) l))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2e-208) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((D * ((M_m * 0.5) / d)) * ((D * (h / d)) / ((2.0 / M_m) * l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m * d) <= 2d-208) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((d * ((m_m * 0.5d0) / d_1)) * ((d * (h / d_1)) / ((2.0d0 / m_m) * l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2e-208) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D * ((M_m * 0.5) / d)) * ((D * (h / d)) / ((2.0 / M_m) * l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (M_m * D) <= 2e-208:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((D * ((M_m * 0.5) / d)) * ((D * (h / d)) / ((2.0 / M_m) * l)))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 2e-208)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D * Float64(Float64(M_m * 0.5) / d)) * Float64(Float64(D * Float64(h / d)) / Float64(Float64(2.0 / M_m) * l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((M_m * D) <= 2e-208)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((D * ((M_m * 0.5) / d)) * ((D * (h / d)) / ((2.0 / M_m) * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 2e-208], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / M$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.0000000000000002e-208

   1. Initial program 85.7%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in M around 0 78.2%

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

   if 2.0000000000000002e-208 < (*.f64 M D)

   1. Initial program 74.3%

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

   3. Simplified 76.5%

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

      1. unpow2 76.5%

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

      2. associate-*r/ 73.2%

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

      3. clear-num 73.2%

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

      4. un-div-inv 73.2%

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

      5. *-un-lft-identity 73.2%

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

      6. times-frac 73.2%

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

      7. metadata-eval 73.2%

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

      8. times-frac 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. frac-times 84.5%

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

      2. associate-*r/ 84.5%

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

   8. Applied egg-rr 84.5%

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

      1. clear-num 84.6%

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

      2. inv-pow 84.6%

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

   10. Applied egg-rr 84.6%

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

       1. unpow-1 84.6%

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

   12. Simplified 84.6%

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

       1. associate-/l* 84.5%

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

       2. clear-num 84.5%

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

       3. *-commutative 84.5%

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

       4. associate-*l* 77.9%

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

       5. associate-/l* 77.8%

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

       6. *-un-lft-identity 77.8%

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

       7. associate-/l* 77.9%

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

       8. associate-/l* 77.9%

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

       9. associate-*l* 77.9%

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

       10. *-commutative 77.9%

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

       11. *-commutative 77.9%

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

   14. Applied egg-rr 77.9%

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

       1. *-lft-identity 77.9%

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

       2. associate-*r/ 75.7%

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

       3. rem-cube-cbrt 75.7%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(M \cdot \color{blue}{{\left(\sqrt[3]{0.5}\right)}^{3}}\right) \cdot D}{d} \cdot \frac{h}{\frac{d}{D} \cdot \left(\frac{2}{M} \cdot \ell\right)}} \]

       4. *-commutative 75.7%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{D \cdot \left(M \cdot {\left(\sqrt[3]{0.5}\right)}^{3}\right)}}{d} \cdot \frac{h}{\frac{d}{D} \cdot \left(\frac{2}{M} \cdot \ell\right)}} \]

       5. associate-/l* 74.6%

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot \frac{M \cdot {\left(\sqrt[3]{0.5}\right)}^{3}}{d}\right)} \cdot \frac{h}{\frac{d}{D} \cdot \left(\frac{2}{M} \cdot \ell\right)}} \]

       6. rem-cube-cbrt 74.5%

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

       7. associate-/r* 73.3%

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

       8. associate-/r/ 74.4%

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

       9. *-commutative 74.4%

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

   16. Simplified 74.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-208}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{D \cdot \frac{h}{d}}{\frac{2}{M} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.6 \cdot 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\frac{w0}{h} - \frac{0.125 \cdot \frac{w0 \cdot \left(\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)\right)}{{d}^{2}}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 9.6e-13)
   w0
   (*
    h
    (-
     (/ w0 h)
     (/ (* 0.125 (/ (* w0 (* (* M_m D) (* M_m D))) (pow d 2.0))) l)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 9.6e-13) {
		tmp = w0;
	} else {
		tmp = h * ((w0 / h) - ((0.125 * ((w0 * ((M_m * D) * (M_m * D))) / pow(d, 2.0))) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 9.6d-13) then
        tmp = w0
    else
        tmp = h * ((w0 / h) - ((0.125d0 * ((w0 * ((m_m * d) * (m_m * d))) / (d_1 ** 2.0d0))) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 9.6e-13) {
		tmp = w0;
	} else {
		tmp = h * ((w0 / h) - ((0.125 * ((w0 * ((M_m * D) * (M_m * D))) / Math.pow(d, 2.0))) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 9.6e-13:
		tmp = w0
	else:
		tmp = h * ((w0 / h) - ((0.125 * ((w0 * ((M_m * D) * (M_m * D))) / math.pow(d, 2.0))) / l))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 9.6e-13)
		tmp = w0;
	else
		tmp = Float64(h * Float64(Float64(w0 / h) - Float64(Float64(0.125 * Float64(Float64(w0 * Float64(Float64(M_m * D) * Float64(M_m * D))) / (d ^ 2.0))) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 9.6e-13)
		tmp = w0;
	else
		tmp = h * ((w0 / h) - ((0.125 * ((w0 * ((M_m * D) * (M_m * D))) / (d ^ 2.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 9.6e-13], w0, N[(h * N[(N[(w0 / h), $MachinePrecision] - N[(N[(0.125 * N[(N[(w0 * N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.5999999999999995e-13

   1. Initial program 82.7%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in M around 0 78.5%

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

   if 9.5999999999999995e-13 < M

   1. Initial program 79.0%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in M around 0 47.4%

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

      1. +-commutative 47.4%

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

      2. *-commutative 47.4%

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

      3. fma-define 47.4%

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

      4. associate-*r* 49.0%

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

      5. *-commutative 49.0%

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

      6. unpow2 49.0%

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

      7. unpow2 49.0%

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

      8. swap-sqr 61.0%

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

      9. unpow2 61.0%

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

      10. *-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in h around -inf 37.9%

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

      1. mul-1-neg 37.9%

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

      2. distribute-rgt-neg-in 37.9%

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

      3. +-commutative 37.9%

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

      4. mul-1-neg 37.9%

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

      5. unsub-neg 37.9%

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

   10. Simplified 48.4%

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

       1. unpow2 48.4%

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

       2. *-commutative 48.4%

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

       3. *-commutative 48.4%

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

   12. Applied egg-rr 48.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.6 \cdot 10^{-13}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\frac{w0}{h} - \frac{0.125 \cdot \frac{w0 \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{{d}^{2}}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.05 \cdot 10^{-12}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\frac{w0}{h} - \frac{0.125 \cdot \left(\left(\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)\right) \cdot \left(w0 \cdot {d}^{-2}\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 1.05e-12)
   w0
   (*
    h
    (-
     (/ w0 h)
     (/ (* 0.125 (* (* (* M_m D) (* M_m D)) (* w0 (pow d -2.0)))) l)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.05e-12) {
		tmp = w0;
	} else {
		tmp = h * ((w0 / h) - ((0.125 * (((M_m * D) * (M_m * D)) * (w0 * pow(d, -2.0)))) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 1.05d-12) then
        tmp = w0
    else
        tmp = h * ((w0 / h) - ((0.125d0 * (((m_m * d) * (m_m * d)) * (w0 * (d_1 ** (-2.0d0))))) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.05e-12) {
		tmp = w0;
	} else {
		tmp = h * ((w0 / h) - ((0.125 * (((M_m * D) * (M_m * D)) * (w0 * Math.pow(d, -2.0)))) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 1.05e-12:
		tmp = w0
	else:
		tmp = h * ((w0 / h) - ((0.125 * (((M_m * D) * (M_m * D)) * (w0 * math.pow(d, -2.0)))) / l))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 1.05e-12)
		tmp = w0;
	else
		tmp = Float64(h * Float64(Float64(w0 / h) - Float64(Float64(0.125 * Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) * Float64(w0 * (d ^ -2.0)))) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.05e-12)
		tmp = w0;
	else
		tmp = h * ((w0 / h) - ((0.125 * (((M_m * D) * (M_m * D)) * (w0 * (d ^ -2.0)))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 1.05e-12], w0, N[(h * N[(N[(w0 / h), $MachinePrecision] - N[(N[(0.125 * N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.04999999999999997e-12

   1. Initial program 82.7%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in M around 0 78.5%

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

   if 1.04999999999999997e-12 < M

   1. Initial program 79.0%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in M around 0 47.4%

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

      1. +-commutative 47.4%

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

      2. *-commutative 47.4%

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

      3. fma-define 47.4%

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

      4. associate-*r* 49.0%

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

      5. *-commutative 49.0%

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

      6. unpow2 49.0%

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

      7. unpow2 49.0%

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

      8. swap-sqr 61.0%

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

      9. unpow2 61.0%

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

      10. *-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in h around -inf 37.9%

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

      1. mul-1-neg 37.9%

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

      2. distribute-rgt-neg-in 37.9%

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

      3. +-commutative 37.9%

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

      4. mul-1-neg 37.9%

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

      5. unsub-neg 37.9%

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

   10. Simplified 48.4%

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

       1. div-inv 48.4%

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

       2. *-commutative 48.4%

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

       3. pow-flip 48.4%

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

       4. metadata-eval 48.4%

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

   12. Applied egg-rr 48.4%

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

       1. associate-*l* 48.4%

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

       2. *-commutative 48.4%

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

   14. Simplified 48.4%

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

       1. unpow2 48.4%

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

       2. *-commutative 48.4%

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

       3. *-commutative 48.4%

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

   16. Applied egg-rr 48.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-12}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;h \cdot \left(\frac{w0}{h} - \frac{0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot {d}^{-2}\right)\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 4e-5)
   w0
   (* -0.125 (/ (* (* (* M_m D) (* M_m D)) (* w0 h)) (* l (pow d 2.0))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M_m * D) * (M_m * D)) * (w0 * h)) / (l * pow(d, 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 4d-5) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((m_m * d) * (m_m * d)) * (w0 * h)) / (l * (d_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M_m * D) * (M_m * D)) * (w0 * h)) / (l * Math.pow(d, 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 4e-5:
		tmp = w0
	else:
		tmp = -0.125 * ((((M_m * D) * (M_m * D)) * (w0 * h)) / (l * math.pow(d, 2.0)))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) * Float64(w0 * h)) / Float64(l * (d ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = -0.125 * ((((M_m * D) * (M_m * D)) * (w0 * h)) / (l * (d ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 4e-5], w0, N[(-0.125 * N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000033e-5

   1. Initial program 82.8%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in M around 0 78.1%

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

   if 4.00000000000000033e-5 < M

   1. Initial program 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in M around 0 46.5%

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

      1. +-commutative 46.5%

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

      2. *-commutative 46.5%

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

      3. fma-define 46.5%

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

      4. associate-*r* 48.2%

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

      5. *-commutative 48.2%

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

      6. unpow2 48.2%

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

      7. unpow2 48.2%

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

      8. swap-sqr 60.3%

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

      9. unpow2 60.3%

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

      10. *-commutative 60.3%

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

   7. Simplified 60.3%

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

   8. Taylor expanded in D around inf 22.0%

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

      1. associate-*r* 22.0%

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

      2. unpow2 22.0%

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

      3. unpow2 22.0%

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

      4. swap-sqr 23.1%

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

      5. unpow2 23.1%

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

      6. *-commutative 23.1%

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

   10. Simplified 23.1%

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

       1. unpow2 47.5%

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

       2. *-commutative 47.5%

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

       3. *-commutative 47.5%

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

   12. Applied egg-rr 23.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \frac{D \cdot 0.5}{d}\right)}{\left(\frac{2}{M\_m} \cdot \frac{d}{D}\right) \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (/ (* h (* M_m (/ (* D 0.5) d))) (* (* (/ 2.0 M_m) (/ d D)) l))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / (((2.0 / M_m) * (d / D)) * l))));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((h * (m_m * ((d * 0.5d0) / d_1))) / (((2.0d0 / m_m) * (d_1 / d)) * l))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / (((2.0 / M_m) * (d / D)) * l))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / (((2.0 / M_m) * (d / D)) * l))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(Float64(D * 0.5) / d))) / Float64(Float64(Float64(2.0 / M_m) * Float64(d / D)) * l)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / (((2.0 / M_m) * (d / D)) * l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 / M$95$m), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*r/ 81.4%

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

   3. clear-num 81.4%

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

   4. un-div-inv 81.4%

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

   5. *-un-lft-identity 81.4%

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

   6. times-frac 81.4%

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

   7. metadata-eval 81.4%

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

   8. times-frac 82.9%

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

6. Applied egg-rr 82.9%

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

   1. frac-times 90.2%

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

   2. associate-*r/ 90.2%

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

8. Applied egg-rr 90.2%

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

9. Final simplification 90.2%

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

Alternative 7: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \frac{D \cdot 0.5}{d}\right)}{\frac{d}{D} \cdot \left(\frac{2}{M\_m} \cdot \ell\right)}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (/ (* h (* M_m (/ (* D 0.5) d))) (* (/ d D) (* (/ 2.0 M_m) l)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / ((d / D) * ((2.0 / M_m) * l)))));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((h * (m_m * ((d * 0.5d0) / d_1))) / ((d_1 / d) * ((2.0d0 / m_m) * l)))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / ((d / D) * ((2.0 / M_m) * l)))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / ((d / D) * ((2.0 / M_m) * l)))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(Float64(D * 0.5) / d))) / Float64(Float64(d / D) * Float64(Float64(2.0 / M_m) * l))))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D * 0.5) / d))) / ((d / D) * ((2.0 / M_m) * l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] * N[(N[(2.0 / M$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*r/ 81.4%

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

   3. clear-num 81.4%

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

   4. un-div-inv 81.4%

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

   5. *-un-lft-identity 81.4%

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

   6. times-frac 81.4%

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

   7. metadata-eval 81.4%

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

   8. times-frac 82.9%

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

6. Applied egg-rr 82.9%

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

   1. frac-times 90.2%

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

   2. associate-*r/ 90.2%

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

8. Applied egg-rr 90.2%

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

   1. pow1 90.2%

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

   2. *-commutative 90.2%

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

10. Applied egg-rr 90.2%

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

    1. unpow1 90.2%

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

    2. associate-*r* 86.5%

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

12. Simplified 86.5%

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

13. Final simplification 86.5%

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

Alternative 8: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \left(D \cdot \frac{M\_m \cdot h}{d}\right)}{\left(\frac{2}{M\_m} \cdot \frac{d}{D}\right) \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (- 1.0 (/ (* 0.5 (* D (/ (* M_m h) d))) (* (* (/ 2.0 M_m) (/ d D)) l))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((0.5 * (D * ((M_m * h) / d))) / (((2.0 / M_m) * (d / D)) * l))));
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - ((0.5d0 * (d * ((m_m * h) / d_1))) / (((2.0d0 / m_m) * (d_1 / d)) * l))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((0.5 * (D * ((M_m * h) / d))) / (((2.0 / M_m) * (d / D)) * l))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((0.5 * (D * ((M_m * h) / d))) / (((2.0 / M_m) * (d / D)) * l))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D * Float64(Float64(M_m * h) / d))) / Float64(Float64(Float64(2.0 / M_m) * Float64(d / D)) * l)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((0.5 * (D * ((M_m * h) / d))) / (((2.0 / M_m) * (d / D)) * l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D * N[(N[(M$95$m * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 / M$95$m), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

   1. unpow2 82.9%

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

   2. associate-*r/ 81.4%

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

   3. clear-num 81.4%

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

   4. un-div-inv 81.4%

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

   5. *-un-lft-identity 81.4%

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

   6. times-frac 81.4%

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

   7. metadata-eval 81.4%

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

   8. times-frac 82.9%

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

6. Applied egg-rr 82.9%

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

   1. frac-times 90.2%

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

   2. associate-*r/ 90.2%

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

8. Applied egg-rr 90.2%

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

9. Taylor expanded in M around 0 85.0%

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

    1. associate-/l* 84.8%

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

11. Simplified 84.8%

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

Alternative 9: 67.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{w0 \cdot h}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 3.1e-62) w0 (/ (* w0 h) h)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.1e-62) {
		tmp = w0;
	} else {
		tmp = (w0 * h) / h;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 3.1d-62) then
        tmp = w0
    else
        tmp = (w0 * h) / h
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.1e-62) {
		tmp = w0;
	} else {
		tmp = (w0 * h) / h;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 3.1e-62:
		tmp = w0
	else:
		tmp = (w0 * h) / h
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 3.1e-62)
		tmp = w0;
	else
		tmp = Float64(Float64(w0 * h) / h);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 3.1e-62)
		tmp = w0;
	else
		tmp = (w0 * h) / h;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 3.1e-62], w0, N[(N[(w0 * h), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.0999999999999999e-62

   1. Initial program 82.2%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in M around 0 78.8%

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

   if 3.0999999999999999e-62 < M

   1. Initial program 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in M around 0 48.7%

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

      1. +-commutative 48.7%

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

      2. *-commutative 48.7%

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

      3. fma-define 48.7%

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

      4. associate-*r* 50.1%

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

      5. *-commutative 50.1%

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

      6. unpow2 50.1%

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

      7. unpow2 50.1%

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

      8. swap-sqr 60.2%

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

      9. unpow2 60.2%

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

      10. *-commutative 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in h around inf 38.0%

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

      1. fma-define 38.0%

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

      2. associate-*r* 38.2%

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

      3. unpow2 38.2%

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

      4. unpow2 38.2%

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

      5. swap-sqr 46.1%

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

      6. unpow2 46.1%

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

   10. Simplified 46.1%

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

   11. Taylor expanded in D around 0 36.5%

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

       1. associate-*r/ 49.3%

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

   13. Applied egg-rr 49.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{w0 \cdot h}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 216.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d) :precision binary64 w0)

C

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

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.8%

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

3. Simplified 82.9%

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

5. Taylor expanded in M around 0 71.0%

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

Reproduce

herbie shell --seed 2024138 
(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))))))