Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 87.8%

Time: 22.0s

Alternatives: 10

Speedup: 1.0×

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: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M\_m \cdot D\_m\right)}^{2} \cdot \left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(h \cdot {M\_m}^{2}\right) \cdot {\left(\frac{D\_m}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 (<= (* M_m D_m) 1e-57)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l))))
   (if (<= (* M_m D_m) 5e+150)
     (*
      w0
      (sqrt
       (+
        1.0
        (* -0.25 (* (pow (* M_m D_m) 2.0) (* (/ h d) (/ (/ 1.0 d) l)))))))
     (*
      w0
      (sqrt
       (- 1.0 (* 0.25 (/ (* (* h (pow M_m 2.0)) (pow (/ D_m d) 2.0)) l))))))))

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 ((M_m * D_m) <= 1e-57) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else if ((M_m * D_m) <= 5e+150) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((h * pow(M_m, 2.0)) * pow((D_m / d), 2.0)) / l))));
	}
	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 ((m_m * d_m) <= 1d-57) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
    else if ((m_m * d_m) <= 5d+150) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m_m * d_m) ** 2.0d0) * ((h / d) * ((1.0d0 / d) / l))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((h * (m_m ** 2.0d0)) * ((d_m / d) ** 2.0d0)) / l))))
    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 ((M_m * D_m) <= 1e-57) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else if ((M_m * D_m) <= 5e+150) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (Math.pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((h * Math.pow(M_m, 2.0)) * Math.pow((D_m / d), 2.0)) / l))));
	}
	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 (M_m * D_m) <= 1e-57:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))
	elif (M_m * D_m) <= 5e+150:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (math.pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((h * math.pow(M_m, 2.0)) * math.pow((D_m / d), 2.0)) / l))))
	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 (Float64(M_m * D_m) <= 1e-57)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))));
	elseif (Float64(M_m * D_m) <= 5e+150)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64((Float64(M_m * D_m) ^ 2.0) * Float64(Float64(h / d) * Float64(Float64(1.0 / d) / l)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(h * (M_m ^ 2.0)) * (Float64(D_m / d) ^ 2.0)) / l)))));
	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 ((M_m * D_m) <= 1e-57)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
	elseif ((M_m * D_m) <= 5e+150)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M_m * D_m) ^ 2.0) * ((h / d) * ((1.0 / d) / l))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((h * (M_m ^ 2.0)) * ((D_m / d) ^ 2.0)) / l))));
	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[N[(M$95$m * D$95$m), $MachinePrecision], 1e-57], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+150], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(1.0 / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(D$95$m / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.99999999999999955e-58

   1. Initial program 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 88.1%

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

      2. add-sqr-sqrt 88.1%

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

      3. pow2 88.1%

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

      4. unpow2 88.1%

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

      5. sqrt-prod 59.2%

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

      6. add-sqr-sqrt 88.1%

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

      7. *-un-lft-identity 88.1%

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

      8. times-frac 88.1%

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

      9. metadata-eval 88.1%

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

   6. Applied egg-rr 88.1%

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

   7. Taylor expanded in M around 0 88.6%

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

   if 9.99999999999999955e-58 < (*.f64 M D) < 5.00000000000000009e150

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in M around 0 36.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 36.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      2. add-sqr-sqrt 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\sqrt{{d}^{2} \cdot \ell} \cdot \sqrt{{d}^{2} \cdot \ell}}}} \]

      3. times-frac 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{\sqrt{{d}^{2} \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)}} \]

      4. sqrt-prod 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      5. unpow2 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      6. sqrt-prod 5.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      7. add-sqr-sqrt 8.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{d} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      8. associate-*r* 11.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      9. pow-prod-down 29.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      10. sqrt-prod 29.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}}\right)} \]

      11. unpow2 29.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}}\right)} \]

      12. sqrt-prod 17.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}}\right)} \]

      13. add-sqr-sqrt 41.7%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{d} \cdot \sqrt{\ell}}\right)} \]

   7. Applied egg-rr 41.7%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ 41.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{1 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}{d \cdot \sqrt{\ell}}}} \]

      2. *-lft-identity 41.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      3. associate-/l* 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      4. associate-*l/ 47.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d \cdot \sqrt{\ell}}\right)}} \]

      5. associate-/r* 47.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \color{blue}{\frac{\frac{h}{d}}{\sqrt{\ell}}}\right)} \]

   9. Simplified 47.2%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 47.2%

          \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right)} \]

       2. cancel-sign-sub-inv 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{\color{blue}{1 + \left(-0.25\right) \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}}\right) \]

       3. metadata-eval 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + \color{blue}{-0.25} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right) \]

       4. frac-times 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}}\right) \]

   11. Applied egg-rr 47.2%

       \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 47.2%

          \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       2. associate-/r* 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}{\sqrt{\ell}}}} \]

       3. associate-*l/ 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d}}}{\sqrt{\ell}}} \]

       4. associate-*l/ 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}}{\sqrt{\ell}}} \]

       5. associate-/r* 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       6. associate-/l* 50.1%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}\right)}} \]

       7. associate-*l* 50.1%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\color{blue}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}\right)} \]

   13. Simplified 50.1%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right)}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 91.5%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \color{blue}{\ell}}\right)} \]

       2. div-inv 91.6%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\color{blue}{h \cdot \frac{1}{d}}}{d \cdot \ell}\right)} \]

       3. times-frac 91.6%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)}\right)} \]

   15. Applied egg-rr 91.6%

       \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)}\right)} \]

   if 5.00000000000000009e150 < (*.f64 M D)

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in M around 0 42.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. times-frac 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}} \]

      2. unpow2 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      3. unpow2 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      4. frac-times 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      5. pow2 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      6. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right)} \]

   7. Applied egg-rr 50.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{h \cdot {M}^{2}}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}} \]

      2. associate-*l/ 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{\left(h \cdot {M}^{2}\right) \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}}} \]

      3. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right)} \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}} \]

   9. Simplified 50.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{\left({M}^{2} \cdot h\right) \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(h \cdot {M}^{2}\right) \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left({M\_m}^{2} \cdot \frac{h}{\ell \cdot {d}^{2}}\right)\right) - -2 \cdot \log D\_m\right)}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) 2e+298)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l))))
   (*
    w0
    (pow
     (pow
      (exp 0.25)
      (-
       (log (* -0.25 (* (pow M_m 2.0) (/ h (* l (pow d 2.0))))))
       (* -2.0 (log 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 (pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 2e+298) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * pow(pow(exp(0.25), (log((-0.25 * (pow(M_m, 2.0) * (h / (l * pow(d, 2.0)))))) - (-2.0 * log(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 ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) <= 2d+298) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * ((exp(0.25d0) ** (log(((-0.25d0) * ((m_m ** 2.0d0) * (h / (l * (d ** 2.0d0)))))) - ((-2.0d0) * log(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 (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 2e+298) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.pow(Math.pow(Math.exp(0.25), (Math.log((-0.25 * (Math.pow(M_m, 2.0) * (h / (l * Math.pow(d, 2.0)))))) - (-2.0 * Math.log(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 math.pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 2e+298:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))
	else:
		tmp = w0 * math.pow(math.pow(math.exp(0.25), (math.log((-0.25 * (math.pow(M_m, 2.0) * (h / (l * math.pow(d, 2.0)))))) - (-2.0 * math.log(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 ((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 2e+298)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * ((exp(0.25) ^ Float64(log(Float64(-0.25 * Float64((M_m ^ 2.0) * Float64(h / Float64(l * (d ^ 2.0)))))) - Float64(-2.0 * log(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 ((((M_m * D_m) / (2.0 * d)) ^ 2.0) <= 2e+298)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
	else
		tmp = w0 * ((exp(0.25) ^ (log((-0.25 * ((M_m ^ 2.0) * (h / (l * (d ^ 2.0)))))) - (-2.0 * log(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[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+298], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[Power[M$95$m, 2.0], $MachinePrecision] * N[(h / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 1.9999999999999999e298

   1. Initial program 87.4%

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

   3. Simplified 86.9%

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

      1. associate-*r/ 95.9%

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

      2. add-sqr-sqrt 95.9%

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

      3. pow2 95.9%

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

      4. unpow2 95.9%

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

      5. sqrt-prod 62.7%

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

      6. add-sqr-sqrt 95.9%

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

      7. *-un-lft-identity 95.9%

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

      8. times-frac 95.9%

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

      9. metadata-eval 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in M around 0 96.9%

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

   if 1.9999999999999999e298 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 44.8%

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

   3. Simplified 46.2%

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

      1. add-sqr-sqrt 46.2%

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

      2. pow2 46.2%

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

   6. Applied egg-rr 46.2%

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in D around inf 23.3%

      \[\leadsto w0 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{2} \]
   8. Step-by-step derivation

      1. exp-prod 23.2%

         \[\leadsto w0 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{2} \]

      2. distribute-lft-neg-in 23.2%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      3. metadata-eval 23.2%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      4. associate-/l* 23.0%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      5. log-rec 23.0%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) + -2 \cdot \color{blue}{\left(-\log D\right)}\right)}\right)}^{2} \]

   9. Simplified 23.0%

      \[\leadsto w0 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right) + -2 \cdot \left(-\log D\right)\right)}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left({M}^{2} \cdot \frac{h}{\ell \cdot {d}^{2}}\right)\right) - -2 \cdot \log D\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.6× speedup

Math

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

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 (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) INFINITY)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M_m (* 2.0 (/ d D_m))) 2.0)))))
   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) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= ((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M_m / (2.0 * (d / D_m))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 ((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M_m / (2.0 * (d / D_m))), 2.0))));
	} else {
		tmp = w0;
	}
	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 (1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= math.inf:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M_m / (2.0 * (d / D_m))), 2.0))))
	else:
		tmp = w0
	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 (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= Inf)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M_m / Float64(2.0 * Float64(d / D_m))) ^ 2.0)))));
	else
		tmp = w0;
	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 ((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))) <= Inf)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m / (2.0 * (d / D_m))) ^ 2.0))));
	else
		tmp = w0;
	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[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m / N[(2.0 * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

   1. Initial program 84.0%

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

   3. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.4%

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

      3. associate-/l* 84.4%

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

   6. Applied egg-rr 84.4%

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

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in M around 0 67.4%

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

4. Final simplification 82.9%

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

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M\_m \cdot D\_m\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) 5e+230)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (pow (* M_m D_m) 2.0) (/ (/ h d) (* d l)))))))))

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 (pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 5e+230) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (pow((M_m * D_m), 2.0) * ((h / d) / (d * l))))));
	}
	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 ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) <= 5d+230) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m_m * d_m) ** 2.0d0) * ((h / d) / (d * l))))))
    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 (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 5e+230) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (Math.pow((M_m * D_m), 2.0) * ((h / d) / (d * l))))));
	}
	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 math.pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 5e+230:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (math.pow((M_m * D_m), 2.0) * ((h / d) / (d * l))))))
	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 ((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 5e+230)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64((Float64(M_m * D_m) ^ 2.0) * Float64(Float64(h / d) / Float64(d * l)))))));
	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 ((((M_m * D_m) / (2.0 * d)) ^ 2.0) <= 5e+230)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M_m * D_m) ^ 2.0) * ((h / d) / (d * l))))));
	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[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+230], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / d), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0000000000000003e230

   1. Initial program 87.6%

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

   3. Simplified 87.1%

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

      1. associate-*r/ 96.3%

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

      2. add-sqr-sqrt 96.3%

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

      3. pow2 96.3%

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

      4. unpow2 96.3%

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

      5. sqrt-prod 62.8%

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

      6. add-sqr-sqrt 96.3%

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

      7. *-un-lft-identity 96.3%

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

      8. times-frac 96.3%

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

      9. metadata-eval 96.3%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in M around 0 97.3%

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

   if 5.0000000000000003e230 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 47.3%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in M around 0 44.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 44.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      2. add-sqr-sqrt 17.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\sqrt{{d}^{2} \cdot \ell} \cdot \sqrt{{d}^{2} \cdot \ell}}}} \]

      3. times-frac 17.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{\sqrt{{d}^{2} \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)}} \]

      4. sqrt-prod 17.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      5. unpow2 17.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      6. sqrt-prod 3.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      7. add-sqr-sqrt 3.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{d} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      8. associate-*r* 3.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      9. pow-prod-down 3.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      10. sqrt-prod 3.4%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}}\right)} \]

      11. unpow2 3.4%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}}\right)} \]

      12. sqrt-prod 3.4%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}}\right)} \]

      13. add-sqr-sqrt 26.1%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{d} \cdot \sqrt{\ell}}\right)} \]

   7. Applied egg-rr 26.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ 26.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{1 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}{d \cdot \sqrt{\ell}}}} \]

      2. *-lft-identity 26.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      3. associate-/l* 27.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      4. associate-*l/ 26.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d \cdot \sqrt{\ell}}\right)}} \]

      5. associate-/r* 26.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \color{blue}{\frac{\frac{h}{d}}{\sqrt{\ell}}}\right)} \]

   9. Simplified 26.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 26.1%

          \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right)} \]

       2. cancel-sign-sub-inv 26.1%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{\color{blue}{1 + \left(-0.25\right) \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}}\right) \]

       3. metadata-eval 26.1%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + \color{blue}{-0.25} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right) \]

       4. frac-times 24.7%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}}\right) \]

   11. Applied egg-rr 24.7%

       \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 24.7%

          \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       2. associate-/r* 24.8%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}{\sqrt{\ell}}}} \]

       3. associate-*l/ 24.8%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d}}}{\sqrt{\ell}}} \]

       4. associate-*l/ 24.8%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}}{\sqrt{\ell}}} \]

       5. associate-/r* 24.7%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       6. associate-/l* 27.4%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}\right)}} \]

       7. associate-*l* 27.4%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\color{blue}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}\right)} \]

   13. Simplified 27.4%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right)}} \]

   14. Taylor expanded in l around 0 58.0%

       \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \color{blue}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M\_m \cdot D\_m\right)}^{2} \cdot \left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(h \cdot {M\_m}^{2}\right) \cdot \frac{D\_m}{d \cdot \frac{d}{D\_m}}}{\ell}}\\ \end{array} \end{array} \]

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 (<= (* M_m D_m) 1e-57)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l))))
   (if (<= (* M_m D_m) 5e+150)
     (*
      w0
      (sqrt
       (+
        1.0
        (* -0.25 (* (pow (* M_m D_m) 2.0) (* (/ h d) (/ (/ 1.0 d) l)))))))
     (*
      w0
      (sqrt
       (-
        1.0
        (* 0.25 (/ (* (* h (pow M_m 2.0)) (/ D_m (* d (/ d D_m)))) l))))))))

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 ((M_m * D_m) <= 1e-57) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else if ((M_m * D_m) <= 5e+150) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((h * pow(M_m, 2.0)) * (D_m / (d * (d / D_m)))) / l))));
	}
	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 ((m_m * d_m) <= 1d-57) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
    else if ((m_m * d_m) <= 5d+150) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m_m * d_m) ** 2.0d0) * ((h / d) * ((1.0d0 / d) / l))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((h * (m_m ** 2.0d0)) * (d_m / (d * (d / d_m)))) / l))))
    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 ((M_m * D_m) <= 1e-57) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else if ((M_m * D_m) <= 5e+150) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (Math.pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((h * Math.pow(M_m, 2.0)) * (D_m / (d * (d / D_m)))) / l))));
	}
	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 (M_m * D_m) <= 1e-57:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))
	elif (M_m * D_m) <= 5e+150:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (math.pow((M_m * D_m), 2.0) * ((h / d) * ((1.0 / d) / l))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((h * math.pow(M_m, 2.0)) * (D_m / (d * (d / D_m)))) / l))))
	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 (Float64(M_m * D_m) <= 1e-57)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))));
	elseif (Float64(M_m * D_m) <= 5e+150)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64((Float64(M_m * D_m) ^ 2.0) * Float64(Float64(h / d) * Float64(Float64(1.0 / d) / l)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(h * (M_m ^ 2.0)) * Float64(D_m / Float64(d * Float64(d / D_m)))) / l)))));
	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 ((M_m * D_m) <= 1e-57)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
	elseif ((M_m * D_m) <= 5e+150)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M_m * D_m) ^ 2.0) * ((h / d) * ((1.0 / d) / l))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((h * (M_m ^ 2.0)) * (D_m / (d * (d / D_m)))) / l))));
	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[N[(M$95$m * D$95$m), $MachinePrecision], 1e-57], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+150], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(1.0 / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d * N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.99999999999999955e-58

   1. Initial program 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 88.1%

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

      2. add-sqr-sqrt 88.1%

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

      3. pow2 88.1%

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

      4. unpow2 88.1%

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

      5. sqrt-prod 59.2%

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

      6. add-sqr-sqrt 88.1%

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

      7. *-un-lft-identity 88.1%

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

      8. times-frac 88.1%

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

      9. metadata-eval 88.1%

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

   6. Applied egg-rr 88.1%

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

   7. Taylor expanded in M around 0 88.6%

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

   if 9.99999999999999955e-58 < (*.f64 M D) < 5.00000000000000009e150

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in M around 0 36.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 36.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      2. add-sqr-sqrt 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\sqrt{{d}^{2} \cdot \ell} \cdot \sqrt{{d}^{2} \cdot \ell}}}} \]

      3. times-frac 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{\sqrt{{d}^{2} \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)}} \]

      4. sqrt-prod 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      5. unpow2 14.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      6. sqrt-prod 5.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      7. add-sqr-sqrt 8.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{d} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      8. associate-*r* 11.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      9. pow-prod-down 29.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      10. sqrt-prod 29.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}}\right)} \]

      11. unpow2 29.6%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}}\right)} \]

      12. sqrt-prod 17.9%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}}\right)} \]

      13. add-sqr-sqrt 41.7%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{d} \cdot \sqrt{\ell}}\right)} \]

   7. Applied egg-rr 41.7%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ 41.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{1 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}{d \cdot \sqrt{\ell}}}} \]

      2. *-lft-identity 41.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      3. associate-/l* 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      4. associate-*l/ 47.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d \cdot \sqrt{\ell}}\right)}} \]

      5. associate-/r* 47.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \color{blue}{\frac{\frac{h}{d}}{\sqrt{\ell}}}\right)} \]

   9. Simplified 47.2%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 47.2%

          \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right)} \]

       2. cancel-sign-sub-inv 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{\color{blue}{1 + \left(-0.25\right) \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}}\right) \]

       3. metadata-eval 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + \color{blue}{-0.25} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right) \]

       4. frac-times 47.2%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}}\right) \]

   11. Applied egg-rr 47.2%

       \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 47.2%

          \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       2. associate-/r* 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}{\sqrt{\ell}}}} \]

       3. associate-*l/ 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d}}}{\sqrt{\ell}}} \]

       4. associate-*l/ 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}}{\sqrt{\ell}}} \]

       5. associate-/r* 47.2%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       6. associate-/l* 50.1%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}\right)}} \]

       7. associate-*l* 50.1%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\color{blue}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}\right)} \]

   13. Simplified 50.1%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right)}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 91.5%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \color{blue}{\ell}}\right)} \]

       2. div-inv 91.6%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\color{blue}{h \cdot \frac{1}{d}}}{d \cdot \ell}\right)} \]

       3. times-frac 91.6%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)}\right)} \]

   15. Applied egg-rr 91.6%

       \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \color{blue}{\left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)}\right)} \]

   if 5.00000000000000009e150 < (*.f64 M D)

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in M around 0 42.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. times-frac 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}} \]

      2. unpow2 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      3. unpow2 42.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      4. frac-times 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      5. pow2 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \]

      6. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right)} \]

   7. Applied egg-rr 50.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{h \cdot {M}^{2}}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right)}} \]

      2. associate-*l/ 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{\left(h \cdot {M}^{2}\right) \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}}} \]

      3. *-commutative 50.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right)} \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}} \]

   9. Simplified 50.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{\left({M}^{2} \cdot h\right) \cdot {\left(\frac{D}{d}\right)}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. unpow2 50.1%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}}{\ell}} \]

       2. clear-num 50.1%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \left(\color{blue}{\frac{1}{\frac{d}{D}}} \cdot \frac{D}{d}\right)}{\ell}} \]

       3. frac-times 49.8%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \color{blue}{\frac{1 \cdot D}{\frac{d}{D} \cdot d}}}{\ell}} \]

       4. *-un-lft-identity 49.8%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \frac{\color{blue}{D}}{\frac{d}{D} \cdot d}}{\ell}} \]

   11. Applied egg-rr 49.8%

       \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \color{blue}{\frac{D}{\frac{d}{D} \cdot d}}}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \left(\frac{h}{d} \cdot \frac{\frac{1}{d}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(h \cdot {M}^{2}\right) \cdot \frac{D}{d \cdot \frac{d}{D}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d} \leq 2 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(M\_m \cdot D\_m\right)}^{2} \cdot \frac{h}{d}}{d \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= (/ (* M_m D_m) (* 2.0 d)) 2e+115)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (/ (* (pow (* M_m D_m) 2.0) (/ h d)) (* d l))))))))

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 (((M_m * D_m) / (2.0 * d)) <= 2e+115) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((pow((M_m * D_m), 2.0) * (h / d)) / (d * l)))));
	}
	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 (((m_m * d_m) / (2.0d0 * d)) <= 2d+115) then
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((((m_m * d_m) ** 2.0d0) * (h / d)) / (d * l)))))
    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 (((M_m * D_m) / (2.0 * d)) <= 2e+115) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((Math.pow((M_m * D_m), 2.0) * (h / d)) / (d * l)))));
	}
	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 ((M_m * D_m) / (2.0 * d)) <= 2e+115:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((math.pow((M_m * D_m), 2.0) * (h / d)) / (d * l)))))
	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 (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) <= 2e+115)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64((Float64(M_m * D_m) ^ 2.0) * Float64(h / d)) / Float64(d * l))))));
	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 (((M_m * D_m) / (2.0 * d)) <= 2e+115)
		tmp = w0 * sqrt((1.0 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((((M_m * D_m) ^ 2.0) * (h / d)) / (d * l)))));
	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[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e+115], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 2e115

   1. Initial program 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 89.9%

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

      2. add-sqr-sqrt 89.9%

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

      3. pow2 89.9%

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

      4. unpow2 89.9%

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

      5. sqrt-prod 53.5%

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

      6. add-sqr-sqrt 89.9%

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

      7. *-un-lft-identity 89.9%

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

      8. times-frac 89.9%

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

      9. metadata-eval 89.9%

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

   6. Applied egg-rr 89.9%

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

   7. Taylor expanded in M around 0 90.7%

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

   if 2e115 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in M around 0 42.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 42.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      2. add-sqr-sqrt 11.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{1 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\sqrt{{d}^{2} \cdot \ell} \cdot \sqrt{{d}^{2} \cdot \ell}}}} \]

      3. times-frac 11.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{\sqrt{{d}^{2} \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)}} \]

      4. sqrt-prod 11.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      5. unpow2 11.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      6. sqrt-prod 0.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      7. add-sqr-sqrt 0.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{\color{blue}{d} \cdot \sqrt{\ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      8. associate-*r* 0.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      9. pow-prod-down 0.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\sqrt{{d}^{2} \cdot \ell}}\right)} \]

      10. sqrt-prod 0.7%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\sqrt{{d}^{2}} \cdot \sqrt{\ell}}}\right)} \]

      11. unpow2 0.7%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\sqrt{\color{blue}{d \cdot d}} \cdot \sqrt{\ell}}\right)} \]

      12. sqrt-prod 0.7%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)} \cdot \sqrt{\ell}}\right)} \]

      13. add-sqr-sqrt 21.5%

          \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{d} \cdot \sqrt{\ell}}\right)} \]

   7. Applied egg-rr 21.5%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{1}{d \cdot \sqrt{\ell}} \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ 21.5%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{1 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}{d \cdot \sqrt{\ell}}}} \]

      2. *-lft-identity 21.5%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      3. associate-/l* 21.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d \cdot \sqrt{\ell}}}}{d \cdot \sqrt{\ell}}} \]

      4. associate-*l/ 21.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d \cdot \sqrt{\ell}}\right)}} \]

      5. associate-/r* 21.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \color{blue}{\frac{\frac{h}{d}}{\sqrt{\ell}}}\right)} \]

   9. Simplified 21.4%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 21.4%

          \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right)} \]

       2. cancel-sign-sub-inv 21.4%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{\color{blue}{1 + \left(-0.25\right) \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}}\right) \]

       3. metadata-eval 21.4%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + \color{blue}{-0.25} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{\frac{h}{d}}{\sqrt{\ell}}\right)}\right) \]

       4. frac-times 19.0%

          \[\leadsto w0 \cdot \left(1 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}}\right) \]

   11. Applied egg-rr 19.0%

       \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 19.0%

          \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       2. associate-/r* 19.0%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}{\sqrt{\ell}}}} \]

       3. associate-*l/ 19.0%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \sqrt{\ell}} \cdot \frac{h}{d}}}{\sqrt{\ell}}} \]

       4. associate-*l/ 19.0%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \sqrt{\ell}}}}{\sqrt{\ell}}} \]

       5. associate-/r* 19.0%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}}} \]

       6. associate-/l* 21.4%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\left(d \cdot \sqrt{\ell}\right) \cdot \sqrt{\ell}}\right)}} \]

       7. associate-*l* 21.4%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{\color{blue}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}\right)} \]

   13. Simplified 21.4%

       \[\leadsto w0 \cdot \color{blue}{\sqrt{1 + -0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{\frac{h}{d}}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r/ 19.0%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}} \]

       2. add-sqr-sqrt 55.8%

          \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \color{blue}{\ell}}} \]

   15. Applied egg-rr 55.8%

       \[\leadsto w0 \cdot \sqrt{1 + -0.25 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{h}{d}}{d \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot \frac{h}{d}}{d \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M\_m \cdot \frac{D\_m}{2 \cdot d}\right)}^{2}}\\ \end{array} \end{array} \]

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_m 6.8e-43)
   w0
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* M_m (/ D_m (* 2.0 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_m <= 6.8e-43) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M_m * (D_m / (2.0 * 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_m <= 6.8d-43) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m_m * (d_m / (2.0d0 * 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_m <= 6.8e-43) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M_m * (D_m / (2.0 * 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_m <= 6.8e-43:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M_m * (D_m / (2.0 * 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_m <= 6.8e-43)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M_m * Float64(D_m / Float64(2.0 * 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_m <= 6.8e-43)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * (D_m / (2.0 * 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$95$m, 6.8e-43], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.8000000000000001e-43

   1. Initial program 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in M around 0 68.6%

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

   if 6.8000000000000001e-43 < D

   1. Initial program 74.3%

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

   3. Simplified 74.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 1.0× speedup

Math

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

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 (/ (* (pow (* 0.5 (/ (* M_m D_m) d)) 2.0) h) l)))))

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 - ((pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
}

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 - ((((0.5d0 * ((m_m * d_m) / d)) ** 2.0d0) * h) / l)))
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 - ((Math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)));
}

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 - ((math.pow((0.5 * ((M_m * D_m) / d)), 2.0) * h) / l)))

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((Float64(0.5 * Float64(Float64(M_m * D_m) / d)) ^ 2.0) * h) / l))))
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 - ((((0.5 * ((M_m * D_m) / d)) ^ 2.0) * h) / l)));
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[(N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

3. Simplified 76.4%

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

   1. associate-*r/ 83.2%

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

   2. add-sqr-sqrt 83.2%

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

   3. pow2 83.2%

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

   4. unpow2 83.2%

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

   5. sqrt-prod 52.4%

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

   6. add-sqr-sqrt 83.2%

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

   7. *-un-lft-identity 83.2%

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

   8. times-frac 83.2%

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

   9. metadata-eval 83.2%

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

6. Applied egg-rr 83.2%

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

7. Taylor expanded in M around 0 84.0%

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

8. Final simplification 84.0%

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

Alternative 9: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 170000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

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 (<= M_m 170000.0) w0 (log (exp 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) {
	double tmp;
	if (M_m <= 170000.0) {
		tmp = w0;
	} else {
		tmp = log(exp(w0));
	}
	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 (m_m <= 170000.0d0) then
        tmp = w0
    else
        tmp = log(exp(w0))
    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 (M_m <= 170000.0) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(w0));
	}
	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 M_m <= 170000.0:
		tmp = w0
	else:
		tmp = math.log(math.exp(w0))
	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 (M_m <= 170000.0)
		tmp = w0;
	else
		tmp = log(exp(w0));
	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 (M_m <= 170000.0)
		tmp = w0;
	else
		tmp = log(exp(w0));
	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[M$95$m, 170000.0], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.7e5

   1. Initial program 78.2%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in M around 0 67.7%

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

   if 1.7e5 < M

   1. Initial program 69.9%

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

   3. Simplified 71.6%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-unprod 27.1%

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

      3. *-commutative 27.1%

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

      4. *-commutative 27.1%

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

      5. swap-sqr 26.9%

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in M around 0 24.2%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   8. Step-by-step derivation

      1. sqrt-pow1 49.7%

         \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

      2. metadata-eval 49.7%

         \[\leadsto {w0}^{\color{blue}{1}} \]

      3. pow1 49.7%

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

      4. add-log-exp 28.3%

         \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   9. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 170000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% 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 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in M around 0 63.8%

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

6. Final simplification 63.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(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))))))