Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 85.8%

Time: 20.4s

Alternatives: 9

Speedup: 1.7×

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: 80.7% 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: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{\frac{\frac{0.25}{d} \cdot \left(\left(M \cdot h\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{d}}}}\\ \mathbf{elif}\;M \cdot D \leq 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\ell \cdot \frac{d}{\frac{h}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-29)
   (*
    w0
    (sqrt
     (+ 1.0 (/ -1.0 (/ l (/ (* (/ 0.25 d) (* (* M h) (* D (* M D)))) d))))))
   (if (<= (* M D) 1e+139)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ (* D (* M (* M D))) (* l (/ d (/ h d))))))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-29) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (((0.25 / d) * ((M * h) * (D * (M * D)))) / d)))));
	} else if ((M * D) <= 1e+139) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((m * d) <= 1d-29) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (((0.25d0 / d_1) * ((m * h) * (d * (m * d)))) / d_1)))))
    else if ((m * d) <= 1d+139) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d * (m * (m * d))) / (l * (d_1 / (h / d_1)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-29) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (((0.25 / d) * ((M * h) * (D * (M * D)))) / d)))));
	} else if ((M * D) <= 1e+139) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M * D) <= 1e-29:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (((0.25 / d) * ((M * h) * (D * (M * D)))) / d)))))
	elif (M * D) <= 1e+139:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-29)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(Float64(Float64(0.25 / d) * Float64(Float64(M * h) * Float64(D * Float64(M * D)))) / d))))));
	elseif (Float64(M * D) <= 1e+139)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * Float64(M * Float64(M * D))) / Float64(l * Float64(d / Float64(h / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M * D) <= 1e-29)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (((0.25 / d) * ((M * h) * (D * (M * D)))) / d)))));
	elseif ((M * D) <= 1e+139)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-29], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(N[(N[(0.25 / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 1e+139], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.99999999999999943e-30

   1. Initial program 87.1%

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

      1. associate-*r/ 92.4%

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

      2. clear-num 92.4%

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

      3. div-inv 92.4%

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

      4. associate-*l* 91.9%

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

      5. associate-/r* 91.9%

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

      6. metadata-eval 91.9%

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

   3. Applied egg-rr 91.9%

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

   4. Taylor expanded in M around 0 64.6%

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

      1. unpow2 64.6%

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

      2. associate-*r/ 64.6%

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

      3. times-frac 74.2%

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

      4. unpow2 74.2%

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

      5. unpow2 74.2%

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

      6. associate-*r* 77.9%

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

      7. associate-*r* 81.1%

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

      8. *-commutative 81.1%

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

      9. associate-*l* 87.3%

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

   6. Simplified 87.3%

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

      1. associate-*r/ 87.3%

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

      2. *-commutative 87.3%

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

   8. Applied egg-rr 87.3%

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

   if 9.99999999999999943e-30 < (*.f64 M D) < 1.00000000000000003e139

   1. Initial program 67.5%

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

   2. Taylor expanded in M around 0 35.5%

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

      1. associate-*r* 35.5%

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

      2. *-commutative 35.5%

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

      3. unpow2 35.5%

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

      4. unpow2 35.5%

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

      5. swap-sqr 70.1%

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

      6. associate-/l* 74.2%

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

      7. *-commutative 74.2%

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

      8. *-commutative 74.2%

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

      9. swap-sqr 35.5%

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

      10. unpow2 35.5%

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

   4. Simplified 35.5%

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

   5. Taylor expanded in d around 0 35.5%

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

      1. unpow2 35.5%

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

      2. associate-*l/ 35.5%

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

      3. *-commutative 35.5%

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

      4. associate-/l* 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in D around 0 38.1%

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

      1. unpow2 38.1%

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

      2. unpow2 38.1%

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

      3. unswap-sqr 81.2%

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

      4. associate-*l* 68.7%

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

   10. Simplified 68.7%

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

   if 1.00000000000000003e139 < (*.f64 M D)

   1. Initial program 59.1%

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

      1. clear-num 59.1%

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

      2. un-div-inv 59.1%

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

      3. div-inv 59.1%

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

      4. associate-*l* 59.1%

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

      5. associate-/r* 59.1%

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

      6. metadata-eval 59.1%

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

   3. Applied egg-rr 59.1%

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

      1. associate-/r/ 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in M around 0 48.0%

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

      1. times-frac 48.3%

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

      2. unpow2 48.3%

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

      3. unpow2 48.3%

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

      4. times-frac 62.9%

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

      5. unpow2 62.9%

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

      6. associate-/l* 62.7%

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

   8. Simplified 62.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{\frac{\frac{0.25}{d} \cdot \left(\left(M \cdot h\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)\right)}{d}}}}\\ \mathbf{elif}\;M \cdot D \leq 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\ell \cdot \frac{d}{\frac{h}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \]

Alternative 2: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-177}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\ell \cdot \frac{d}{\frac{h}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.25e-177)
   w0
   (if (<= M 1.95e-20)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ (* D (* M (* M D))) (* l (/ d (/ h d))))))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-177) {
		tmp = w0;
	} else if (M <= 1.95e-20) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.25d-177) then
        tmp = w0
    else if (m <= 1.95d-20) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d * (m * (m * d))) / (l * (d_1 / (h / d_1)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e-177) {
		tmp = w0;
	} else if (M <= 1.95e-20) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.25e-177:
		tmp = w0
	elif M <= 1.95e-20:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.25e-177)
		tmp = w0;
	elseif (M <= 1.95e-20)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * Float64(M * Float64(M * D))) / Float64(l * Float64(d / Float64(h / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.25e-177)
		tmp = w0;
	elseif (M <= 1.95e-20)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (M * (M * D))) / (l * (d / (h / d)))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.25e-177], w0, If[LessEqual[M, 1.95e-20], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.25e-177

   1. Initial program 88.0%

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

   2. Taylor expanded in M around 0 73.9%

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

   if 1.25e-177 < M < 1.95000000000000004e-20

   1. Initial program 81.5%

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

   2. Taylor expanded in M around 0 62.3%

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

      1. associate-*r* 65.6%

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

      2. *-commutative 65.6%

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

      3. unpow2 65.6%

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

      4. unpow2 65.6%

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

      5. swap-sqr 78.7%

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

      6. associate-/l* 75.5%

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

      7. *-commutative 75.5%

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

      8. *-commutative 75.5%

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

      9. swap-sqr 62.3%

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

      10. unpow2 62.3%

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

   4. Simplified 62.3%

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

   5. Taylor expanded in d around 0 62.3%

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

      1. unpow2 62.3%

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

      2. associate-*l/ 65.6%

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

      3. *-commutative 65.6%

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

      4. associate-/l* 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in D around 0 65.6%

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

      1. unpow2 65.6%

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

      2. unpow2 65.6%

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

      3. unswap-sqr 81.3%

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

      4. associate-*l* 81.4%

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

   10. Simplified 81.4%

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

   if 1.95000000000000004e-20 < M

   1. Initial program 63.3%

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

      1. clear-num 63.3%

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

      2. un-div-inv 63.3%

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

      3. div-inv 63.3%

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

      4. associate-*l* 63.3%

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

      5. associate-/r* 63.3%

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

      6. metadata-eval 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. associate-/r/ 68.2%

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

   5. Applied egg-rr 68.2%

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

   6. Taylor expanded in M around 0 45.6%

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

      1. times-frac 42.5%

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

      2. unpow2 42.5%

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

      3. unpow2 42.5%

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

      4. times-frac 53.9%

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

      5. unpow2 53.9%

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

      6. associate-/l* 54.8%

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

   8. Simplified 54.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-177}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}{\ell \cdot \frac{d}{\frac{h}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \]

Alternative 3: 80.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{\frac{0.25}{d} \cdot \frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.8e-78)
   (*
    w0
    (sqrt
     (+ 1.0 (/ -1.0 (/ l (* (/ 0.25 d) (/ (* (* M h) (* D (* M D))) d)))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.8e-78) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / ((0.25 / d) * (((M * h) * (D * (M * D))) / d))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.8d-78) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / ((0.25d0 / d_1) * (((m * h) * (d * (m * d))) / d_1))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.8e-78) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / ((0.25 / d) * (((M * h) * (D * (M * D))) / d))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.8e-78:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / ((0.25 / d) * (((M * h) * (D * (M * D))) / d))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.8e-78)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(Float64(0.25 / d) * Float64(Float64(Float64(M * h) * Float64(D * Float64(M * D))) / d)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.8e-78)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / ((0.25 / d) * (((M * h) * (D * (M * D))) / d))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.8e-78], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(N[(0.25 / d), $MachinePrecision] * N[(N[(N[(M * h), $MachinePrecision] * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8000000000000001e-78

   1. Initial program 87.0%

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

      1. associate-*r/ 91.4%

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

      2. clear-num 91.4%

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

      3. div-inv 91.3%

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

      4. associate-*l* 90.8%

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

      5. associate-/r* 90.8%

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

      6. metadata-eval 90.8%

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

   3. Applied egg-rr 90.8%

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

   4. Taylor expanded in M around 0 61.5%

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

      1. unpow2 61.5%

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

      2. associate-*r/ 61.5%

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

      3. times-frac 70.1%

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

      4. unpow2 70.1%

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

      5. unpow2 70.1%

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

      6. associate-*r* 74.0%

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

      7. associate-*r* 78.0%

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

      8. *-commutative 78.0%

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

      9. associate-*l* 87.7%

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

   6. Simplified 87.7%

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

   if 1.8000000000000001e-78 < M

   1. Initial program 67.2%

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

      1. clear-num 67.2%

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

      2. un-div-inv 67.2%

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

      3. div-inv 67.2%

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

      4. associate-*l* 67.2%

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

      5. associate-/r* 67.2%

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

      6. metadata-eval 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-/r/ 71.2%

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

   5. Applied egg-rr 71.2%

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

   6. Taylor expanded in M around 0 51.2%

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

      1. times-frac 48.6%

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

      2. unpow2 48.6%

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

      3. unpow2 48.6%

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

      4. times-frac 59.4%

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

      5. unpow2 59.4%

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

      6. associate-/l* 60.2%

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

   8. Simplified 60.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{\frac{0.25}{d} \cdot \frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{\frac{M}{\ell}}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.1e-77)
   (* w0 (+ 1.0 (* (* D (* D (* (/ M d) (/ (/ M l) (/ d h))))) -0.125)))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M M) (/ l h)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.1e-77) {
		tmp = w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.1d-77) then
        tmp = w0 * (1.0d0 + ((d * (d * ((m / d_1) * ((m / l) / (d_1 / h))))) * (-0.125d0)))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * m) / (l / h))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.1e-77) {
		tmp = w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.1e-77:
		tmp = w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.1e-77)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(M / l) / Float64(d / h))))) * -0.125)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * M) / Float64(l / h)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.1e-77)
		tmp = w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * M) / (l / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.1e-77], N[(w0 * N[(1.0 + N[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.10000000000000003e-77

   1. Initial program 87.1%

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

   2. Taylor expanded in M around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. times-frac 57.8%

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

      3. unpow2 57.8%

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

      4. unpow2 57.8%

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

      5. unpow2 57.8%

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

   4. Simplified 57.8%

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

   5. Taylor expanded in D around 0 58.4%

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

      1. unpow2 58.4%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

      5. times-frac 59.4%

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

      6. associate-/r/ 58.3%

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

      7. *-rgt-identity 58.3%

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

      8. associate-*r/ 57.7%

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

      9. unpow2 57.7%

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

      10. associate-*l* 57.2%

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

      11. associate-*l* 64.8%

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

      12. associate-*r/ 65.4%

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

      13. *-commutative 65.4%

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

      14. *-lft-identity 65.4%

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

      15. unpow2 65.4%

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

      16. associate-/r/ 65.2%

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

      17. *-commutative 65.2%

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

      18. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

      1. times-frac 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in M around 0 65.7%

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

       1. associate-/r* 68.5%

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

       2. unpow2 68.5%

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

       3. associate-/l* 67.4%

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

       4. unpow2 67.4%

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

       5. associate-*l/ 70.9%

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

       6. associate-/r* 67.4%

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

       7. associate-/l/ 69.2%

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

       8. associate-*r/ 72.0%

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

       9. *-commutative 72.0%

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

       10. times-frac 79.2%

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

   12. Simplified 79.2%

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

   if 1.10000000000000003e-77 < M

   1. Initial program 66.8%

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

      1. clear-num 66.7%

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

      2. un-div-inv 66.7%

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

      3. div-inv 66.7%

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

      4. associate-*l* 66.7%

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

      5. associate-/r* 66.7%

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

      6. metadata-eval 66.7%

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

   3. Applied egg-rr 66.7%

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

      1. associate-/r/ 70.9%

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

   5. Applied egg-rr 70.9%

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

   6. Taylor expanded in M around 0 50.6%

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

      1. times-frac 47.9%

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

      2. unpow2 47.9%

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

      3. unpow2 47.9%

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

      4. times-frac 58.8%

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

      5. unpow2 58.8%

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

      6. associate-/l* 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{\frac{M}{\ell}}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\\ \end{array} \]

Alternative 5: 78.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(D \cdot \left(\frac{M}{\ell} \cdot \frac{\frac{M}{d}}{\frac{d}{h}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 5.2e+80)
   (* w0 (+ 1.0 (* -0.125 (* D (* D (* (/ M l) (/ (/ M d) (/ d h))))))))
   (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ D d) (/ D d)) (* M (* M h))) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.2e+80) {
		tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 5.2d+80) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * (d * ((m / l) * ((m / d_1) / (d_1 / h)))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((d / d_1) * (d / d_1)) * (m * (m * h))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.2e+80) {
		tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.2e+80:
		tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.2e+80)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(D * Float64(Float64(M / l) * Float64(Float64(M / d) / Float64(d / h))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M * h))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.2e+80)
		tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 5.2e+80], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(D * N[(N[(M / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.19999999999999963e80

   1. Initial program 80.2%

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

   2. Taylor expanded in M around 0 53.7%

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

      1. *-commutative 53.7%

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

      2. times-frac 52.7%

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

      3. unpow2 52.7%

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

      4. unpow2 52.7%

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

      5. unpow2 52.7%

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

   4. Simplified 52.7%

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

   5. Taylor expanded in D around 0 53.7%

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

      1. unpow2 53.7%

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

      2. associate-*r* 54.7%

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

      3. unpow2 54.7%

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

      4. unpow2 54.7%

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

      5. times-frac 54.2%

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

      6. associate-/r/ 53.5%

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

      7. *-rgt-identity 53.5%

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

      8. associate-*r/ 53.0%

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

      9. unpow2 53.0%

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

      10. associate-*l* 52.6%

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

      11. associate-*l* 58.4%

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

      12. associate-*r/ 58.9%

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

      13. *-commutative 58.9%

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

      14. *-lft-identity 58.9%

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

      15. unpow2 58.9%

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

      16. associate-/r/ 58.8%

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

      17. *-commutative 58.8%

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

      18. associate-/l* 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in M around 0 59.3%

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

      1. associate-/r* 61.8%

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

      2. unpow2 61.8%

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

      3. unpow2 61.8%

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

      4. times-frac 66.8%

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

      5. associate-/r/ 63.8%

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

      6. associate-*r/ 70.9%

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

      7. associate-*l/ 68.8%

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

      8. associate-/r/ 68.8%

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

      9. *-commutative 68.8%

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

      10. associate-/r* 71.8%

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

   10. Simplified 71.8%

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

   if 5.19999999999999963e80 < d

   1. Initial program 84.8%

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

   2. Taylor expanded in M around 0 63.6%

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

      1. *-commutative 63.6%

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

      2. times-frac 61.7%

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

      3. unpow2 61.7%

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

      4. unpow2 61.7%

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

      5. unpow2 61.7%

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

   4. Simplified 61.7%

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

      1. associate-*r/ 63.7%

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

      2. associate-/l* 74.7%

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

      3. associate-*l* 80.4%

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in D around 0 69.4%

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

      1. unpow2 69.4%

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

      2. unpow2 69.4%

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

      3. times-frac 81.3%

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

   9. Simplified 81.3%

      \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell} \cdot -0.125\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(D \cdot \left(\frac{M}{\ell} \cdot \frac{\frac{M}{d}}{\frac{d}{h}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 6: 78.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + \left(D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{\frac{M}{\ell}}{\frac{d}{h}}\right)\right)\right) \cdot -0.125\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* (* D (* D (* (/ M d) (/ (/ M l) (/ d h))))) -0.125))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((d * (d * ((m / d_1) * ((m / l) / (d_1 / h))))) * (-0.125d0)))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(M / l) / Float64(d / h))))) * -0.125)))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + ((D * (D * ((M / d) * ((M / l) / (d / h))))) * -0.125));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in M around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. times-frac 54.5%

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

   3. unpow2 54.5%

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

   4. unpow2 54.5%

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

   5. unpow2 54.5%

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

4. Simplified 54.5%

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

5. Taylor expanded in D around 0 55.7%

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

   1. unpow2 55.7%

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

   2. associate-*r* 56.9%

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

   3. unpow2 56.9%

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

   4. unpow2 56.9%

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

   5. times-frac 55.3%

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

   6. associate-/r/ 54.0%

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

   7. *-rgt-identity 54.0%

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

   8. associate-*r/ 53.6%

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

   9. unpow2 53.6%

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

   10. associate-*l* 53.6%

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

   11. associate-*l* 60.6%

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

   12. associate-*r/ 60.9%

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

   13. *-commutative 60.9%

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

   14. *-lft-identity 60.9%

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

   15. unpow2 60.9%

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

   16. associate-/r/ 61.7%

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

   17. *-commutative 61.7%

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

   18. associate-/l* 63.7%

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

7. Simplified 63.7%

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

   1. times-frac 71.6%

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

9. Applied egg-rr 71.6%

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

10. Taylor expanded in M around 0 62.4%

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

    1. associate-/r* 64.4%

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

    2. unpow2 64.4%

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

    3. associate-/l* 63.3%

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

    4. unpow2 63.3%

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

    5. associate-*l/ 66.1%

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

    6. associate-/r* 63.7%

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

    7. associate-/l/ 63.4%

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

    8. associate-*r/ 66.1%

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

    9. *-commutative 66.1%

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

    10. times-frac 74.0%

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

12. Simplified 74.0%

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

13. Final simplification 74.0%

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

Alternative 7: 78.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(D \cdot \left(\frac{M}{\ell} \cdot \frac{\frac{M}{d}}{\frac{d}{h}}\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* D (* D (* (/ M l) (/ (/ M d) (/ d h)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((-0.125d0) * (d * (d * ((m / l) * ((m / d_1) / (d_1 / h)))))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(D * Float64(Float64(M / l) * Float64(Float64(M / d) / Float64(d / h))))))))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M / l) * ((M / d) / (d / h)))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(D * N[(N[(M / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in M around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. times-frac 54.5%

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

   3. unpow2 54.5%

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

   4. unpow2 54.5%

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

   5. unpow2 54.5%

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

4. Simplified 54.5%

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

5. Taylor expanded in D around 0 55.7%

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

   1. unpow2 55.7%

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

   2. associate-*r* 56.9%

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

   3. unpow2 56.9%

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

   4. unpow2 56.9%

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

   5. times-frac 55.3%

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

   6. associate-/r/ 54.0%

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

   7. *-rgt-identity 54.0%

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

   8. associate-*r/ 53.6%

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

   9. unpow2 53.6%

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

   10. associate-*l* 53.6%

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

   11. associate-*l* 60.6%

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

   12. associate-*r/ 60.9%

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

   13. *-commutative 60.9%

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

   14. *-lft-identity 60.9%

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

   15. unpow2 60.9%

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

   16. associate-/r/ 61.7%

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

   17. *-commutative 61.7%

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

   18. associate-/l* 63.7%

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

7. Simplified 63.7%

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

8. Taylor expanded in M around 0 62.4%

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

   1. associate-/r* 64.4%

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

   2. unpow2 64.4%

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

   3. unpow2 64.4%

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

   4. times-frac 68.9%

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

   5. associate-/r/ 66.1%

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

   6. associate-*r/ 73.7%

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

   7. associate-*l/ 71.6%

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

   8. associate-/r/ 71.6%

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

   9. *-commutative 71.6%

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

   10. associate-/r* 74.4%

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

10. Simplified 74.4%

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

11. Final simplification 74.4%

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

Alternative 8: 78.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)}{\ell}\right)\right)\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* D (* D (/ (* M (* (/ h d) (/ M d))) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (D * (D * ((M * ((h / d) * (M / d))) / l)))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((-0.125d0) * (d * (d * ((m * ((h / d_1) * (m / d_1))) / l)))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (D * (D * ((M * ((h / d) * (M / d))) / l)))));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * (D * (D * ((M * ((h / d) * (M / d))) / l)))))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(D * Float64(Float64(M * Float64(Float64(h / d) * Float64(M / d))) / l))))))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * (D * (D * ((M * ((h / d) * (M / d))) / l)))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(D * N[(N[(M * N[(N[(h / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in M around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. times-frac 54.5%

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

   3. unpow2 54.5%

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

   4. unpow2 54.5%

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

   5. unpow2 54.5%

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

4. Simplified 54.5%

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

5. Taylor expanded in D around 0 55.7%

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

   1. unpow2 55.7%

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

   2. associate-*r* 56.9%

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

   3. unpow2 56.9%

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

   4. unpow2 56.9%

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

   5. times-frac 55.3%

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

   6. associate-/r/ 54.0%

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

   7. *-rgt-identity 54.0%

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

   8. associate-*r/ 53.6%

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

   9. unpow2 53.6%

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

   10. associate-*l* 53.6%

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

   11. associate-*l* 60.6%

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

   12. associate-*r/ 60.9%

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

   13. *-commutative 60.9%

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

   14. *-lft-identity 60.9%

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

   15. unpow2 60.9%

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

   16. associate-/r/ 61.7%

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

   17. *-commutative 61.7%

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

   18. associate-/l* 63.7%

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

7. Simplified 63.7%

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

   1. times-frac 71.6%

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

9. Applied egg-rr 71.6%

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

    1. associate-*l/ 73.7%

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

    2. associate-/r/ 76.4%

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

11. Applied egg-rr 76.4%

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

12. Final simplification 76.4%

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

Alternative 9: 67.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in M around 0 67.6%

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

3. Final simplification 67.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023279 
(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))))))