Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.4%

Time: 19.6s

Alternatives: 7

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: 81.1% 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: 88.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 85.8%

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

   2. frac-times 86.2%

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

   3. *-commutative 86.2%

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

   4. frac-2neg 86.2%

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

   5. *-commutative 86.2%

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

   6. associate-*l/ 85.8%

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

   7. *-commutative 85.8%

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

   8. div-inv 85.8%

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

   9. associate-/r* 85.8%

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

   10. metadata-eval 85.8%

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

6. Applied egg-rr 85.8%

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

   1. distribute-lft-neg-in 85.8%

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

   2. associate-/l* 86.9%

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

8. Simplified 86.9%

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

   1. neg-mul-1 86.9%

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

   2. unpow2 86.9%

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

   3. times-frac 89.5%

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

   4. associate-*r/ 89.5%

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

   5. associate-*r/ 89.5%

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

10. Applied egg-rr 89.5%

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

11. Taylor expanded in D around 0 88.1%

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

    1. associate-/l* 89.5%

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

    2. associate-*r/ 89.5%

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

13. Simplified 89.5%

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

14. Final simplification 89.5%

    \[\leadsto w0 \cdot \sqrt{1 + \frac{h}{\frac{-1}{D \cdot \frac{M \cdot 0.5}{d}} \cdot \frac{\ell}{\frac{D \cdot 0.5}{\frac{d}{M}}}}} \]
15. Add Preprocessing

Alternative 2: 86.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.6 \cdot 10^{-45}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D_m \cdot \frac{M_m}{\frac{d}{0.5}}\right) \cdot \frac{h}{\frac{\ell}{D_m} \cdot \frac{-2}{\frac{M_m}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(\frac{D_m}{d} \cdot \frac{h \cdot M_m}{\ell}\right)\right) \cdot \frac{D_m \cdot M_m}{d \cdot 2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 5.6e-45)
   (*
    w0
    (sqrt
     (+
      1.0
      (* (* D_m (/ M_m (/ d 0.5))) (/ h (* (/ l D_m) (/ -2.0 (/ M_m d))))))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* 0.5 (* (/ D_m d) (/ (* h M_m) l))) (/ (* D_m M_m) (* d 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 5.6e-45) {
		tmp = w0 * sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.5 * ((D_m / d) * ((h * M_m) / l))) * ((D_m * M_m) / (d * 2.0)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 5.6d-45) then
        tmp = w0 * sqrt((1.0d0 + ((d_m * (m_m / (d / 0.5d0))) * (h / ((l / d_m) * ((-2.0d0) / (m_m / d)))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * ((d_m / d) * ((h * m_m) / l))) * ((d_m * m_m) / (d * 2.0d0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 5.6e-45) {
		tmp = w0 * Math.sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * ((D_m / d) * ((h * M_m) / l))) * ((D_m * M_m) / (d * 2.0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 5.6e-45:
		tmp = w0 * math.sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * ((D_m / d) * ((h * M_m) / l))) * ((D_m * M_m) / (d * 2.0)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 5.6e-45)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m * Float64(M_m / Float64(d / 0.5))) * Float64(h / Float64(Float64(l / D_m) * Float64(-2.0 / Float64(M_m / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(Float64(D_m / d) * Float64(Float64(h * M_m) / l))) * Float64(Float64(D_m * M_m) / Float64(d * 2.0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 5.6e-45)
		tmp = w0 * sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	else
		tmp = w0 * sqrt((1.0 - ((0.5 * ((D_m / d) * ((h * M_m) / l))) * ((D_m * M_m) / (d * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 5.6e-45], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m * N[(M$95$m / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(l / D$95$m), $MachinePrecision] * N[(-2.0 / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.6000000000000003e-45

   1. Initial program 80.4%

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

   3. Simplified 79.8%

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

      1. associate-*r/ 82.7%

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

      2. frac-times 83.3%

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

      3. *-commutative 83.3%

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

      4. frac-2neg 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*l/ 82.7%

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

      7. *-commutative 82.7%

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

      8. div-inv 82.7%

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

      9. associate-/r* 82.7%

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

      10. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

      1. distribute-lft-neg-in 82.7%

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

      2. associate-/l* 84.3%

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

   8. Simplified 84.3%

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

      1. neg-mul-1 84.3%

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

      2. unpow2 84.3%

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

      3. times-frac 88.0%

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

      4. associate-*r/ 88.1%

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

      5. associate-*r/ 88.0%

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

   10. Applied egg-rr 88.0%

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

       1. associate-*l/ 88.1%

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

       2. associate-/r* 86.4%

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

       3. associate-/l* 86.4%

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

       4. neg-mul-1 86.4%

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

       5. associate-/l* 86.4%

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

       6. associate-/r/ 85.4%

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

       7. distribute-neg-frac 85.4%

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

       8. associate-/r/ 85.4%

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

       9. associate-*r/ 83.8%

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

       10. div-inv 83.8%

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

       11. metadata-eval 83.8%

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

   12. Applied egg-rr 83.8%

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

       1. associate-*l/ 84.3%

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

       2. associate-/l* 84.8%

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

       3. neg-mul-1 84.8%

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

       4. *-commutative 84.8%

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

       5. times-frac 84.8%

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

       6. metadata-eval 84.8%

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

       7. *-commutative 84.8%

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

       8. times-frac 86.4%

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

   14. Simplified 86.4%

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

       1. expm1-log1p-u 85.9%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)\right)} \]

       2. expm1-udef 85.9%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)} - 1\right)} \]

   16. Applied egg-rr 82.3%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)} - 1\right)} \]
   17. Step-by-step derivation

       1. expm1-def 82.3%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)\right)} \]

       2. expm1-log1p 82.7%

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

       3. sub-neg 82.7%

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

       4. *-commutative 82.7%

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

       5. distribute-rgt-neg-in 82.7%

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

       6. times-frac 83.8%

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

       7. *-rgt-identity 83.8%

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

       8. times-frac 83.8%

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

       9. associate-*r/ 85.4%

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

       10. metadata-eval 85.4%

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

       11. associate-*l* 85.4%

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

       12. associate-/r/ 85.4%

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

       13. distribute-frac-neg 85.4%

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

       14. remove-double-neg 85.4%

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

   18. Simplified 85.4%

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

   if 5.6000000000000003e-45 < d

   1. Initial program 79.9%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 92.8%

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

      2. frac-times 92.8%

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

      3. *-commutative 92.8%

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

      4. frac-2neg 92.8%

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

      5. *-commutative 92.8%

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

      6. associate-*l/ 92.8%

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

      7. *-commutative 92.8%

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

      8. div-inv 92.8%

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

      9. associate-/r* 92.8%

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

      10. metadata-eval 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. distribute-lft-neg-in 92.8%

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

      2. associate-/l* 92.8%

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

   8. Simplified 92.8%

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

      1. neg-mul-1 92.8%

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

      2. unpow2 92.8%

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

      3. times-frac 92.8%

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

      4. associate-*r/ 92.8%

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

      5. associate-*r/ 92.8%

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

   10. Applied egg-rr 92.8%

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

       1. associate-*l/ 92.8%

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

       2. associate-/r* 87.8%

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

       3. associate-/l* 87.8%

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

       4. neg-mul-1 87.8%

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

       5. associate-/l* 87.8%

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

       6. associate-/r/ 87.8%

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

       7. distribute-neg-frac 87.8%

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

       8. associate-/r/ 87.8%

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

       9. associate-*r/ 86.6%

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

       10. div-inv 86.6%

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

       11. metadata-eval 86.6%

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

   12. Applied egg-rr 86.6%

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

   13. Taylor expanded in h around 0 87.7%

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

       1. times-frac 85.1%

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

   15. Simplified 85.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.6 \cdot 10^{-45}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \frac{h}{\frac{\ell}{D} \cdot \frac{-2}{\frac{M}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{h \cdot M}{\ell}\right)\right) \cdot \frac{D \cdot M}{d \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.2 \cdot 10^{-45}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D_m \cdot \frac{M_m}{\frac{d}{0.5}}\right) \cdot \frac{h}{\frac{\ell}{D_m} \cdot \frac{-2}{\frac{M_m}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h}{\frac{-2 \cdot \left(\frac{d}{D_m} \cdot \frac{\ell}{M_m}\right)}{\frac{M_m}{d} \cdot \frac{D_m}{2}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= d 2.2e-45)
   (*
    w0
    (sqrt
     (+
      1.0
      (* (* D_m (/ M_m (/ d 0.5))) (/ h (* (/ l D_m) (/ -2.0 (/ M_m d))))))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/ h (/ (* -2.0 (* (/ d D_m) (/ l M_m))) (* (/ M_m d) (/ D_m 2.0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2.2e-45) {
		tmp = w0 * sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (h / ((-2.0 * ((d / D_m) * (l / M_m))) / ((M_m / d) * (D_m / 2.0))))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 2.2d-45) then
        tmp = w0 * sqrt((1.0d0 + ((d_m * (m_m / (d / 0.5d0))) * (h / ((l / d_m) * ((-2.0d0) / (m_m / d)))))))
    else
        tmp = w0 * sqrt((1.0d0 + (h / (((-2.0d0) * ((d / d_m) * (l / m_m))) / ((m_m / d) * (d_m / 2.0d0))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 2.2e-45) {
		tmp = w0 * Math.sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (h / ((-2.0 * ((d / D_m) * (l / M_m))) / ((M_m / d) * (D_m / 2.0))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if d <= 2.2e-45:
		tmp = w0 * math.sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (h / ((-2.0 * ((d / D_m) * (l / M_m))) / ((M_m / d) * (D_m / 2.0))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 2.2e-45)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m * Float64(M_m / Float64(d / 0.5))) * Float64(h / Float64(Float64(l / D_m) * Float64(-2.0 / Float64(M_m / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(h / Float64(Float64(-2.0 * Float64(Float64(d / D_m) * Float64(l / M_m))) / Float64(Float64(M_m / d) * Float64(D_m / 2.0)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 2.2e-45)
		tmp = w0 * sqrt((1.0 + ((D_m * (M_m / (d / 0.5))) * (h / ((l / D_m) * (-2.0 / (M_m / d)))))));
	else
		tmp = w0 * sqrt((1.0 + (h / ((-2.0 * ((d / D_m) * (l / M_m))) / ((M_m / d) * (D_m / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 2.2e-45], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m * N[(M$95$m / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(l / D$95$m), $MachinePrecision] * N[(-2.0 / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(h / N[(N[(-2.0 * N[(N[(d / D$95$m), $MachinePrecision] * N[(l / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.19999999999999993e-45

   1. Initial program 80.4%

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

   3. Simplified 79.8%

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

      1. associate-*r/ 82.7%

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

      2. frac-times 83.3%

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

      3. *-commutative 83.3%

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

      4. frac-2neg 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*l/ 82.7%

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

      7. *-commutative 82.7%

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

      8. div-inv 82.7%

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

      9. associate-/r* 82.7%

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

      10. metadata-eval 82.7%

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

   6. Applied egg-rr 82.7%

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

      1. distribute-lft-neg-in 82.7%

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

      2. associate-/l* 84.3%

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

   8. Simplified 84.3%

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

      1. neg-mul-1 84.3%

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

      2. unpow2 84.3%

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

      3. times-frac 88.0%

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

      4. associate-*r/ 88.1%

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

      5. associate-*r/ 88.0%

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

   10. Applied egg-rr 88.0%

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

       1. associate-*l/ 88.1%

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

       2. associate-/r* 86.4%

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

       3. associate-/l* 86.4%

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

       4. neg-mul-1 86.4%

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

       5. associate-/l* 86.4%

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

       6. associate-/r/ 85.4%

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

       7. distribute-neg-frac 85.4%

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

       8. associate-/r/ 85.4%

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

       9. associate-*r/ 83.8%

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

       10. div-inv 83.8%

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

       11. metadata-eval 83.8%

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

   12. Applied egg-rr 83.8%

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

       1. associate-*l/ 84.3%

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

       2. associate-/l* 84.8%

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

       3. neg-mul-1 84.8%

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

       4. *-commutative 84.8%

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

       5. times-frac 84.8%

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

       6. metadata-eval 84.8%

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

       7. *-commutative 84.8%

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

       8. times-frac 86.4%

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

   14. Simplified 86.4%

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

       1. expm1-log1p-u 85.9%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)\right)} \]

       2. expm1-udef 85.9%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)} - 1\right)} \]

   16. Applied egg-rr 82.3%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)} - 1\right)} \]
   17. Step-by-step derivation

       1. expm1-def 82.3%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)\right)} \]

       2. expm1-log1p 82.7%

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

       3. sub-neg 82.7%

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

       4. *-commutative 82.7%

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

       5. distribute-rgt-neg-in 82.7%

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

       6. times-frac 83.8%

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

       7. *-rgt-identity 83.8%

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

       8. times-frac 83.8%

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

       9. associate-*r/ 85.4%

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

       10. metadata-eval 85.4%

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

       11. associate-*l* 85.4%

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

       12. associate-/r/ 85.4%

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

       13. distribute-frac-neg 85.4%

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

       14. remove-double-neg 85.4%

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

   18. Simplified 85.4%

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

   if 2.19999999999999993e-45 < d

   1. Initial program 79.9%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 92.8%

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

      2. frac-times 92.8%

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

      3. *-commutative 92.8%

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

      4. frac-2neg 92.8%

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

      5. *-commutative 92.8%

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

      6. associate-*l/ 92.8%

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

      7. *-commutative 92.8%

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

      8. div-inv 92.8%

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

      9. associate-/r* 92.8%

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

      10. metadata-eval 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. distribute-lft-neg-in 92.8%

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

      2. associate-/l* 92.8%

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

   8. Simplified 92.8%

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

      1. neg-mul-1 92.8%

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

      2. unpow2 92.8%

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

      3. times-frac 92.8%

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

      4. associate-*r/ 92.8%

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

      5. associate-*r/ 92.8%

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

   10. Applied egg-rr 92.8%

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

       1. associate-*l/ 92.8%

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

       2. associate-/r* 87.8%

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

       3. associate-/l* 87.8%

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

       4. neg-mul-1 87.8%

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

       5. associate-/l* 87.8%

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

       6. associate-/r/ 87.8%

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

       7. distribute-neg-frac 87.8%

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

       8. associate-/r/ 87.8%

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

       9. associate-*r/ 86.6%

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

       10. div-inv 86.6%

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

       11. metadata-eval 86.6%

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

   12. Applied egg-rr 86.6%

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

       1. associate-*l/ 86.6%

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

       2. associate-/l* 86.6%

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

       3. neg-mul-1 86.6%

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

       4. *-commutative 86.6%

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

       5. times-frac 86.6%

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

       6. metadata-eval 86.6%

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

       7. *-commutative 86.6%

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

       8. times-frac 87.8%

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

   14. Simplified 87.8%

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

   15. Taylor expanded in l around 0 91.6%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{-h}{\frac{-2 \cdot \color{blue}{\frac{d \cdot \ell}{D \cdot M}}}{\frac{M}{d} \cdot \frac{D}{2}}}} \]
   16. Step-by-step derivation

       1. times-frac 89.1%

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

   17. Simplified 89.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.2 \cdot 10^{-45}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \frac{h}{\frac{\ell}{D} \cdot \frac{-2}{\frac{M}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h}{\frac{-2 \cdot \left(\frac{d}{D} \cdot \frac{\ell}{M}\right)}{\frac{M}{d} \cdot \frac{D}{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 1.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ (* M_m 0.5) d))))
   (* w0 (sqrt (+ 1.0 (/ h (* (/ -1.0 t_0) (/ l t_0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M_m * 0.5) / d);
	return w0 * sqrt((1.0 + (h / ((-1.0 / t_0) * (l / t_0)))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d_m * ((m_m * 0.5d0) / d)
    code = w0 * sqrt((1.0d0 + (h / (((-1.0d0) / t_0) * (l / t_0)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M_m * 0.5) / d);
	return w0 * Math.sqrt((1.0 + (h / ((-1.0 / t_0) * (l / t_0)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * ((M_m * 0.5) / d)
	return w0 * math.sqrt((1.0 + (h / ((-1.0 / t_0) * (l / t_0)))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(Float64(M_m * 0.5) / d))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(h / Float64(Float64(-1.0 / t_0) * Float64(l / t_0))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * ((M_m * 0.5) / d);
	tmp = w0 * sqrt((1.0 + (h / ((-1.0 / t_0) * (l / t_0)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(h / N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 85.8%

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

   2. frac-times 86.2%

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

   3. *-commutative 86.2%

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

   4. frac-2neg 86.2%

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

   5. *-commutative 86.2%

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

   6. associate-*l/ 85.8%

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

   7. *-commutative 85.8%

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

   8. div-inv 85.8%

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

   9. associate-/r* 85.8%

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

   10. metadata-eval 85.8%

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

6. Applied egg-rr 85.8%

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

   1. distribute-lft-neg-in 85.8%

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

   2. associate-/l* 86.9%

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

8. Simplified 86.9%

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

   1. neg-mul-1 86.9%

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

   2. unpow2 86.9%

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

   3. times-frac 89.5%

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

   4. associate-*r/ 89.5%

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

   5. associate-*r/ 89.5%

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

10. Applied egg-rr 89.5%

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

11. Final simplification 89.5%

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

Alternative 5: 84.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 85.8%

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

   2. frac-times 86.2%

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

   3. *-commutative 86.2%

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

   4. frac-2neg 86.2%

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

   5. *-commutative 86.2%

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

   6. associate-*l/ 85.8%

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

   7. *-commutative 85.8%

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

   8. div-inv 85.8%

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

   9. associate-/r* 85.8%

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

   10. metadata-eval 85.8%

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

6. Applied egg-rr 85.8%

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

   1. distribute-lft-neg-in 85.8%

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

   2. associate-/l* 86.9%

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

8. Simplified 86.9%

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

   1. neg-mul-1 86.9%

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

   2. unpow2 86.9%

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

   3. times-frac 89.5%

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

   4. associate-*r/ 89.5%

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

   5. associate-*r/ 89.5%

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

10. Applied egg-rr 89.5%

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

    1. associate-*l/ 89.5%

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

    2. associate-/r* 86.8%

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

    3. associate-/l* 86.8%

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

    4. neg-mul-1 86.8%

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

    5. associate-/l* 86.8%

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

    6. associate-/r/ 86.1%

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

    7. distribute-neg-frac 86.1%

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

    8. associate-/r/ 86.1%

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

    9. associate-*r/ 84.7%

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

    10. div-inv 84.7%

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

    11. metadata-eval 84.7%

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

12. Applied egg-rr 84.7%

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

    1. associate-*l/ 85.0%

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

    2. associate-/l* 85.4%

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

    3. neg-mul-1 85.4%

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

    4. *-commutative 85.4%

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

    5. times-frac 85.4%

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

    6. metadata-eval 85.4%

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

    7. *-commutative 85.4%

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

    8. times-frac 86.8%

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

14. Simplified 86.8%

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

    1. expm1-log1p-u 86.3%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)\right)} \]

    2. expm1-udef 86.3%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\frac{-2 \cdot \frac{\frac{\ell}{D}}{\frac{M}{d}}}{\frac{M}{d} \cdot \frac{D}{2}}}}\right)} - 1\right)} \]

16. Applied egg-rr 83.8%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)} - 1\right)} \]
17. Step-by-step derivation

    1. expm1-def 83.8%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{-h}{\left(\frac{\ell}{D} \cdot \frac{d}{M}\right) \cdot -2} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}\right)\right)} \]

    2. expm1-log1p 84.2%

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

    3. sub-neg 84.2%

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

    4. *-commutative 84.2%

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

    5. distribute-rgt-neg-in 84.2%

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

    6. times-frac 84.7%

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

    7. *-rgt-identity 84.7%

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

    8. times-frac 84.7%

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

    9. associate-*r/ 86.1%

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

    10. metadata-eval 86.1%

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

    11. associate-*l* 86.1%

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

    12. associate-/r/ 86.1%

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

    13. distribute-frac-neg 86.1%

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

    14. remove-double-neg 86.1%

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

18. Simplified 86.1%

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

19. Final simplification 86.1%

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

Alternative 6: 86.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Simplified 79.9%

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

   1. associate-*r/ 85.8%

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

   2. frac-times 86.2%

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

   3. *-commutative 86.2%

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

   4. frac-2neg 86.2%

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

   5. *-commutative 86.2%

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

   6. associate-*l/ 85.8%

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

   7. *-commutative 85.8%

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

   8. div-inv 85.8%

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

   9. associate-/r* 85.8%

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

   10. metadata-eval 85.8%

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

6. Applied egg-rr 85.8%

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

   1. distribute-lft-neg-in 85.8%

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

   2. associate-/l* 86.9%

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

8. Simplified 86.9%

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

   1. neg-mul-1 86.9%

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

   2. unpow2 86.9%

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

   3. times-frac 89.5%

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

   4. associate-*r/ 89.5%

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

   5. associate-*r/ 89.5%

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

10. Applied egg-rr 89.5%

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

    1. associate-*l/ 89.5%

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

    2. associate-/r* 86.8%

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

    3. associate-/l* 86.8%

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

    4. neg-mul-1 86.8%

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

    5. associate-/l* 86.8%

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

    6. associate-/r/ 86.1%

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

    7. distribute-neg-frac 86.1%

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

    8. associate-/r/ 86.1%

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

    9. associate-*r/ 84.7%

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

    10. div-inv 84.7%

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

    11. metadata-eval 84.7%

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

12. Applied egg-rr 84.7%

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

    1. associate-*l/ 85.0%

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

    2. associate-/l* 85.4%

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

    3. neg-mul-1 85.4%

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

    4. *-commutative 85.4%

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

    5. times-frac 85.4%

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

    6. metadata-eval 85.4%

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

    7. *-commutative 85.4%

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

    8. times-frac 86.8%

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

14. Simplified 86.8%

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

    1. *-un-lft-identity 86.8%

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

    2. associate-/r/ 86.1%

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

    3. *-commutative 86.1%

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

    4. div-inv 86.1%

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

    5. clear-num 86.1%

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

    6. frac-times 84.7%

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

    7. *-commutative 84.7%

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

    8. times-frac 84.2%

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

16. Applied egg-rr 84.2%

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

    1. *-commutative 84.2%

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

    2. associate-*r* 84.2%

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

    3. *-commutative 84.2%

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

    4. times-frac 84.7%

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

    5. associate-*l/ 84.7%

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

    6. times-frac 84.7%

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

    7. metadata-eval 84.7%

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

    8. associate-*r/ 86.1%

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

    9. associate-*l* 86.1%

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

    10. associate-/r/ 86.1%

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

    11. neg-mul-1 86.1%

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

    12. *-commutative 86.1%

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

    13. times-frac 86.1%

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

    14. metadata-eval 86.1%

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

    15. times-frac 83.1%

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

    16. *-commutative 83.1%

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

    17. associate-*r/ 86.2%

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

18. Simplified 86.2%

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

19. Final simplification 86.2%

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

Alternative 7: 67.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Simplified 79.9%

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

5. Taylor expanded in D around 0 64.4%

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

6. Final simplification 64.4%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))))