Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.1%

Time: 16.0s

Alternatives: 12

Speedup: 10.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= (/ h l) -2e-30)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* (/ M d) (* D 0.5)) 2.0)))))))
   (* w0 (+ 1.0 (* (/ (* D (* D (* (/ M d) (/ (* h M) d)))) l) -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) {
	double tmp;
	if ((h / l) <= -2e-30) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow(((M / d) * (D * 0.5)), 2.0))))));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	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 ((h / l) <= (-2d-30)) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0))))))
    else
        tmp = w0 * (1.0d0 + (((d * (d * ((m / d_1) * ((h * m) / d_1)))) / l) * (-0.125d0)))
    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 ((h / l) <= -2e-30) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow(((M / d) * (D * 0.5)), 2.0))))));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	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 (h / l) <= -2e-30:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow(((M / d) * (D * 0.5)), 2.0))))))
	else:
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125))
	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(h / l) <= -2e-30)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))) / l) * -0.125)));
	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 ((h / l) <= -2e-30)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * (((M / d) * (D * 0.5)) ^ 2.0))))));
	else
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	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[(h / l), $MachinePrecision], -2e-30], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -2e-30

   1. Initial program 75.0%

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

      1. times-frac 75.0%

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

   3. Simplified 75.0%

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

      1. associate-*r/ 81.2%

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

      2. clear-num 81.2%

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

      3. frac-times 81.2%

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

      4. *-commutative 81.2%

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

      5. frac-times 82.3%

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

      6. div-inv 82.3%

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

      7. metadata-eval 82.3%

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

   5. Applied egg-rr 82.3%

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

   if -2e-30 < (/.f64 h l)

   1. Initial program 88.3%

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

      1. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. *-commutative 58.1%

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

      5. times-frac 58.8%

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

      6. unpow2 58.8%

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

      7. *-commutative 58.8%

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

      8. unpow2 58.8%

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

      9. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-*l/ 60.1%

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

      2. times-frac 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. associate-*l* 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in M around 0 67.9%

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

       1. unpow2 67.9%

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

       2. unpow2 67.9%

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

       3. associate-*r* 71.6%

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

       4. times-frac 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 2: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* M D) (* d 2.0)) 2.0)) 5e-28)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (* w0 (sqrt (- 1.0 (* (/ D d) (* (/ D (/ l (* M (* h M)))) (/ 0.25 d))))))))

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 (((h / l) * pow(((M * D) / (d * 2.0)), 2.0)) <= 5e-28) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 (((h / l) * (((m * d) / (d_1 * 2.0d0)) ** 2.0d0)) <= 5d-28) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * ((d / (l / (m * (h * m)))) * (0.25d0 / d_1)))))
    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 (((h / l) * Math.pow(((M * D) / (d * 2.0)), 2.0)) <= 5e-28) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 ((h / l) * math.pow(((M * D) / (d * 2.0)), 2.0)) <= 5e-28:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))))
	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(Float64(h / l) * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) <= 5e-28)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(Float64(D / Float64(l / Float64(M * Float64(h * M)))) * Float64(0.25 / d))))));
	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 (((h / l) * (((M * D) / (d * 2.0)) ^ 2.0)) <= 5e-28)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e-28], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(N[(D / N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 5.0000000000000002e-28

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. times-frac 89.0%

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

   3. Simplified 89.0%

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

   if 5.0000000000000002e-28 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 10.0%

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

      1. times-frac 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in M around 0 40.8%

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

      1. associate-*r/ 40.8%

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

      2. *-commutative 40.8%

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

      3. *-commutative 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 45.8%

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

      6. unpow2 45.8%

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

      7. unpow2 45.8%

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

      8. swap-sqr 50.8%

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

      9. *-commutative 50.8%

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

      10. *-commutative 50.8%

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

      11. unpow2 50.8%

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

      12. associate-*l* 55.7%

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

   6. Simplified 55.7%

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

   7. Taylor expanded in h around 0 45.7%

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

      1. *-commutative 45.7%

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

      2. unpow2 45.7%

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

      3. associate-*r* 50.7%

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

      4. unpow2 50.7%

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

      5. associate-*l* 55.2%

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

   9. Simplified 55.2%

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

       1. times-frac 60.7%

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

   11. Applied egg-rr 60.7%

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

       1. *-un-lft-identity 60.7%

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

       2. *-commutative 60.7%

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

       3. times-frac 60.6%

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

   13. Applied egg-rr 60.6%

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

       1. *-lft-identity 60.6%

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

       2. associate-*l* 60.6%

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

       3. associate-/l* 65.9%

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

   15. Simplified 65.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-28}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* (/ D d) (* (/ D (/ l (* M (* h M)))) (/ 0.25 d))))))
   (if (<= (/ h l) -2e-30)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
     (* w0 (+ 1.0 (* (/ (* D (* D (* (/ M d) (/ (* h M) d)))) l) -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) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	} else if ((h / l) <= -2e-30) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	return tmp;
}

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 ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	} else if ((h / l) <= -2e-30) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	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 (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))))
	elif (h / l) <= -2e-30:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	else:
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125))
	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(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(Float64(D / Float64(l / Float64(M * Float64(h * M)))) * Float64(0.25 / d))))));
	elseif (Float64(h / l) <= -2e-30)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))) / l) * -0.125)));
	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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	elseif ((h / l) <= -2e-30)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	else
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	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[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(N[(D / N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-30], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 44.7%

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

      1. times-frac 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in M around 0 61.3%

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

      1. associate-*r/ 61.3%

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

      2. *-commutative 61.3%

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

      3. *-commutative 61.3%

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

      4. *-commutative 61.3%

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

      5. associate-*l* 61.5%

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

      6. unpow2 61.5%

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

      7. unpow2 61.5%

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

      8. swap-sqr 66.2%

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

      9. *-commutative 66.2%

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

      10. *-commutative 66.2%

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

      11. unpow2 66.2%

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

      12. associate-*l* 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in h around 0 61.3%

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

      1. *-commutative 61.3%

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

      2. unpow2 61.3%

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

      3. associate-*r* 61.3%

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

      4. unpow2 61.3%

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

      5. associate-*l* 66.2%

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

   9. Simplified 66.2%

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

       1. times-frac 66.2%

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

   11. Applied egg-rr 66.2%

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

       1. *-un-lft-identity 66.2%

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

       2. *-commutative 66.2%

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

       3. times-frac 74.9%

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

   13. Applied egg-rr 74.9%

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

       1. *-lft-identity 74.9%

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

       2. associate-*l* 74.9%

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

       3. associate-/l* 70.8%

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

   15. Simplified 70.8%

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

   if -inf.0 < (/.f64 h l) < -2e-30

   1. Initial program 84.3%

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

      1. times-frac 84.4%

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

   3. Simplified 84.4%

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

   if -2e-30 < (/.f64 h l)

   1. Initial program 88.3%

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

      1. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. *-commutative 58.1%

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

      5. times-frac 58.8%

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

      6. unpow2 58.8%

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

      7. *-commutative 58.8%

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

      8. unpow2 58.8%

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

      9. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-*l/ 60.1%

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

      2. times-frac 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. associate-*l* 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in M around 0 67.9%

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

       1. unpow2 67.9%

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

       2. unpow2 67.9%

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

       3. associate-*r* 71.6%

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

       4. times-frac 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 4: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= (/ h l) -2e-30)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l))))
   (* w0 (+ 1.0 (* (/ (* D (* D (* (/ M d) (/ (* h M) d)))) l) -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) {
	double tmp;
	if ((h / l) <= -2e-30) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	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 ((h / l) <= (-2d-30)) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = w0 * (1.0d0 + (((d * (d * ((m / d_1) * ((h * m) / d_1)))) / l) * (-0.125d0)))
    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 ((h / l) <= -2e-30) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	}
	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 (h / l) <= -2e-30:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))
	else:
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125))
	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(h / l) <= -2e-30)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))) / l) * -0.125)));
	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 ((h / l) <= -2e-30)
		tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
	else
		tmp = w0 * (1.0 + (((D * (D * ((M / d) * ((h * M) / d)))) / l) * -0.125));
	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[(h / l), $MachinePrecision], -2e-30], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -2e-30

   1. Initial program 75.0%

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

      1. times-frac 75.0%

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

   3. Simplified 75.0%

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

      1. frac-times 75.0%

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

      2. associate-*r/ 81.2%

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

      3. *-commutative 81.2%

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

      4. frac-times 82.2%

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

      5. div-inv 82.2%

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

      6. metadata-eval 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -2e-30 < (/.f64 h l)

   1. Initial program 88.3%

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

      1. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. *-commutative 58.1%

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

      5. times-frac 58.8%

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

      6. unpow2 58.8%

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

      7. *-commutative 58.8%

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

      8. unpow2 58.8%

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

      9. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. associate-*l/ 60.1%

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

      2. times-frac 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. associate-*l* 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in M around 0 67.9%

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

       1. unpow2 67.9%

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

       2. unpow2 67.9%

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

       3. associate-*r* 71.6%

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

       4. times-frac 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{-82}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.42 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 7.2e-82)
   (* w0 (+ 1.0 (* -0.125 (/ (* D (* D (/ h (* (/ d M) (/ d M))))) l))))
   (if (<= d 1.42e+115)
     (* w0 (sqrt (- 1.0 (* (/ 0.25 d) (/ (* D (* D (* M (* h M)))) (* l d))))))
     (* w0 (+ 1.0 (* -0.125 (/ (* D (* D (* (/ h d) (/ (* M 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) {
	double tmp;
	if (d <= 7.2e-82) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	} else if (d <= 1.42e+115) {
		tmp = w0 * sqrt((1.0 - ((0.25 / d) * ((D * (D * (M * (h * M)))) / (l * d)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * ((h / d) * ((M * M) / d)))) / 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 <= 7.2d-82) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d * (h / ((d_1 / m) * (d_1 / m))))) / l)))
    else if (d_1 <= 1.42d+115) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / d_1) * ((d * (d * (m * (h * m)))) / (l * d_1)))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d * ((h / d_1) * ((m * m) / d_1)))) / 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 <= 7.2e-82) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	} else if (d <= 1.42e+115) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / d) * ((D * (D * (M * (h * M)))) / (l * d)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * ((h / d) * ((M * M) / d)))) / 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 <= 7.2e-82:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)))
	elif d <= 1.42e+115:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / d) * ((D * (D * (M * (h * M)))) / (l * d)))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * ((h / d) * ((M * M) / d)))) / 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 <= 7.2e-82)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(h / Float64(Float64(d / M) * Float64(d / M))))) / l))));
	elseif (d <= 1.42e+115)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / d) * Float64(Float64(D * Float64(D * Float64(M * Float64(h * M)))) / Float64(l * d))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(Float64(h / d) * Float64(Float64(M * M) / d)))) / 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 <= 7.2e-82)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	elseif (d <= 1.42e+115)
		tmp = w0 * sqrt((1.0 - ((0.25 / d) * ((D * (D * (M * (h * M)))) / (l * d)))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * ((h / d) * ((M * M) / d)))) / 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, 7.2e-82], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.42e+115], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / d), $MachinePrecision] * N[(N[(D * N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D * N[(N[(h / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.19999999999999996e-82

   1. Initial program 81.8%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 48.3%

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

      1. associate-*r/ 48.3%

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

      2. *-commutative 48.3%

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

      3. associate-*r/ 48.3%

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

      4. *-commutative 48.3%

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

      5. times-frac 49.6%

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

      6. unpow2 49.6%

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

      7. *-commutative 49.6%

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

      8. unpow2 49.6%

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

      9. unpow2 49.6%

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

   6. Simplified 49.6%

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

      1. associate-*l/ 49.7%

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

      2. times-frac 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. associate-*l* 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in M around 0 58.6%

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

       1. *-commutative 58.6%

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

       2. associate-/l* 60.7%

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

       3. unpow2 60.7%

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

       4. unpow2 60.7%

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

       5. times-frac 71.6%

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

   13. Simplified 71.6%

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

   if 7.19999999999999996e-82 < d < 1.42e115

   1. Initial program 79.9%

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

      1. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 64.6%

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

      1. associate-*r/ 64.6%

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

      2. *-commutative 64.6%

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

      3. *-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l* 62.8%

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

      6. unpow2 62.8%

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

      7. unpow2 62.8%

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

      8. swap-sqr 74.0%

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

      9. *-commutative 74.0%

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

      10. *-commutative 74.0%

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

      11. unpow2 74.0%

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

      12. associate-*l* 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in h around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. unpow2 64.6%

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

      3. associate-*r* 66.5%

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

      4. unpow2 66.5%

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

      5. associate-*l* 68.4%

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

   9. Simplified 68.4%

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

       1. times-frac 74.2%

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

   11. Applied egg-rr 74.2%

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

   if 1.42e115 < d

   1. Initial program 91.7%

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

      1. times-frac 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in M around 0 62.5%

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

      1. associate-*r/ 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-*r/ 62.5%

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

      4. *-commutative 62.5%

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

      5. times-frac 60.4%

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

      6. unpow2 60.4%

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

      7. *-commutative 60.4%

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

      8. unpow2 60.4%

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

      9. unpow2 60.4%

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

   6. Simplified 60.4%

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

      1. associate-*l/ 66.7%

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

      2. times-frac 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. associate-*l* 81.7%

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

   10. Simplified 81.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{-82}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.42 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-151}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \end{array} \]

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 5e-151)
   (* w0 (+ 1.0 (* -0.125 (/ (* D (* D (/ h (* (/ d M) (/ d M))))) l))))
   (* w0 (sqrt (- 1.0 (* (/ D d) (* (/ D (/ l (* M (* h M)))) (/ 0.25 d))))))))

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 <= 5e-151) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 <= 5d-151) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d * (h / ((d_1 / m) * (d_1 / m))))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * ((d / (l / (m * (h * m)))) * (0.25d0 / d_1)))))
    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 <= 5e-151) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 <= 5e-151:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))))
	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 <= 5e-151)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(h / Float64(Float64(d / M) * Float64(d / M))))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(Float64(D / Float64(l / Float64(M * Float64(h * M)))) * Float64(0.25 / d))))));
	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 <= 5e-151)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (h / ((d / M) * (d / M))))) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	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, 5e-151], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(N[(D / N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.00000000000000003e-151

   1. Initial program 86.8%

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

      1. times-frac 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in M around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-*r/ 59.3%

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

      4. *-commutative 59.3%

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

      5. times-frac 60.5%

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

      6. unpow2 60.5%

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

      7. *-commutative 60.5%

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

      8. unpow2 60.5%

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

      9. unpow2 60.5%

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

   6. Simplified 60.5%

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

      1. associate-*l/ 62.4%

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

      2. times-frac 70.6%

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

   8. Applied egg-rr 70.6%

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

      1. associate-*l* 76.3%

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

   10. Simplified 76.3%

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

   11. Taylor expanded in M around 0 68.3%

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

       1. *-commutative 68.3%

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

       2. associate-/l* 69.6%

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

       3. unpow2 69.6%

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

       4. unpow2 69.6%

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

       5. times-frac 81.3%

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

   13. Simplified 81.3%

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

   if 5.00000000000000003e-151 < M

   1. Initial program 77.0%

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

      1. times-frac 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in M around 0 45.8%

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

      1. associate-*r/ 45.8%

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

      2. *-commutative 45.8%

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

      3. *-commutative 45.8%

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

      4. *-commutative 45.8%

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

      5. associate-*l* 48.0%

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

      6. unpow2 48.0%

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

      7. unpow2 48.0%

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

      8. swap-sqr 64.2%

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

      9. *-commutative 64.2%

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

      10. *-commutative 64.2%

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

      11. unpow2 64.2%

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

      12. associate-*l* 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in h around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. associate-*r* 52.2%

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

      4. unpow2 52.2%

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

      5. associate-*l* 58.8%

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

   9. Simplified 58.8%

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

       1. times-frac 62.1%

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

   11. Applied egg-rr 62.1%

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

       1. *-un-lft-identity 62.1%

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

       2. *-commutative 62.1%

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

       3. times-frac 66.6%

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

   13. Applied egg-rr 66.6%

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

       1. *-lft-identity 66.6%

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

       2. associate-*l* 66.6%

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

       3. associate-/l* 65.3%

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

   15. Simplified 65.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-151}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \]

Alternative 7: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.1 \cdot 10^{-253}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \left(M \cdot \frac{h}{d}\right)\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \end{array} \]

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 4.1e-253)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ (* D D) l) (* M (* M (/ h d))))) d))))
   (* w0 (sqrt (- 1.0 (* (/ D d) (* (/ D (/ l (* M (* h M)))) (/ 0.25 d))))))))

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 <= 4.1e-253) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (((D * D) / l) * (M * (M * (h / d))))) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 <= 4.1d-253) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (((d * d) / l) * (m * (m * (h / d_1))))) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * ((d / (l / (m * (h * m)))) * (0.25d0 / d_1)))))
    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 <= 4.1e-253) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (((D * D) / l) * (M * (M * (h / d))))) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	}
	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 <= 4.1e-253:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (((D * D) / l) * (M * (M * (h / d))))) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))))
	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 <= 4.1e-253)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(D * D) / l) * Float64(M * Float64(M * Float64(h / d))))) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(Float64(D / Float64(l / Float64(M * Float64(h * M)))) * Float64(0.25 / d))))));
	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 <= 4.1e-253)
		tmp = w0 * sqrt((1.0 - ((0.25 * (((D * D) / l) * (M * (M * (h / d))))) / d)));
	else
		tmp = w0 * sqrt((1.0 - ((D / d) * ((D / (l / (M * (h * M)))) * (0.25 / d)))));
	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, 4.1e-253], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(M * N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(N[(D / N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.10000000000000002e-253

   1. Initial program 81.7%

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

      1. times-frac 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in M around 0 51.2%

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

      1. associate-*r/ 51.2%

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

      2. *-commutative 51.2%

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

      3. *-commutative 51.2%

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

      4. *-commutative 51.2%

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

      5. associate-*l* 52.8%

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

      6. unpow2 52.8%

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

      7. unpow2 52.8%

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

      8. swap-sqr 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. unpow2 67.0%

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

      12. associate-*l* 71.6%

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

   6. Simplified 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. times-frac 77.9%

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

      3. pow2 77.9%

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

      4. *-commutative 77.9%

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

   8. Applied egg-rr 77.9%

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

      1. *-lft-identity 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/l* 77.9%

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

      4. associate-*r/ 77.9%

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

      5. *-commutative 77.9%

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

   10. Simplified 77.9%

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

   11. Taylor expanded in h around 0 60.8%

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

       1. associate-*r/ 60.8%

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

       2. *-commutative 60.8%

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

       3. associate-*r/ 60.8%

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

       4. times-frac 63.9%

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

       5. unpow2 63.9%

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

       6. associate-*r/ 64.7%

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

       7. unpow2 64.7%

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

       8. associate-*l* 68.6%

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

   13. Simplified 68.6%

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

   if 4.10000000000000002e-253 < 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. Step-by-step derivation

      1. times-frac 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in M around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. *-commutative 58.9%

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

      3. *-commutative 58.9%

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

      4. *-commutative 58.9%

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

      5. associate-*l* 59.0%

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

      6. unpow2 59.0%

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

      7. unpow2 59.0%

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

      8. swap-sqr 69.1%

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

      9. *-commutative 69.1%

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

      10. *-commutative 69.1%

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

      11. unpow2 69.1%

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

      12. associate-*l* 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in h around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. unpow2 60.5%

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

      3. associate-*r* 64.4%

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

      4. unpow2 64.4%

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

      5. associate-*l* 66.8%

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

   9. Simplified 66.8%

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

       1. times-frac 77.1%

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

   11. Applied egg-rr 77.1%

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

       1. *-un-lft-identity 77.1%

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

       2. *-commutative 77.1%

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

       3. times-frac 80.4%

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

   13. Applied egg-rr 80.4%

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

       1. *-lft-identity 80.4%

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

       2. associate-*l* 80.4%

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

       3. associate-/l* 78.5%

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

   15. Simplified 78.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.1 \cdot 10^{-253}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \left(M \cdot \frac{h}{d}\right)\right)\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{D}{\frac{\ell}{M \cdot \left(h \cdot M\right)}} \cdot \frac{0.25}{d}\right)}\\ \end{array} \]

Alternative 8: 70.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.9 \cdot 10^{-147}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 12500000 \lor \neg \left(M \leq 1.3 \cdot 10^{+38}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 3.9e-147)
   w0
   (if (or (<= M 12500000.0) (not (<= M 1.3e+38)))
     (* w0 (+ 1.0 (* -0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d))))))
     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) {
	double tmp;
	if (M <= 3.9e-147) {
		tmp = w0;
	} else if ((M <= 12500000.0) || !(M <= 1.3e+38)) {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	} else {
		tmp = w0;
	}
	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 <= 3.9d-147) then
        tmp = w0
    else if ((m <= 12500000.0d0) .or. (.not. (m <= 1.3d+38))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * d) / l) * ((h * (m * m)) / (d_1 * d_1)))))
    else
        tmp = w0
    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 <= 3.9e-147) {
		tmp = w0;
	} else if ((M <= 12500000.0) || !(M <= 1.3e+38)) {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	} else {
		tmp = w0;
	}
	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 <= 3.9e-147:
		tmp = w0
	elif (M <= 12500000.0) or not (M <= 1.3e+38):
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))))
	else:
		tmp = w0
	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 <= 3.9e-147)
		tmp = w0;
	elseif ((M <= 12500000.0) || !(M <= 1.3e+38))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))))));
	else
		tmp = w0;
	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 <= 3.9e-147)
		tmp = w0;
	elseif ((M <= 12500000.0) || ~((M <= 1.3e+38)))
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	else
		tmp = w0;
	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, 3.9e-147], w0, If[Or[LessEqual[M, 12500000.0], N[Not[LessEqual[M, 1.3e+38]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if M < 3.8999999999999998e-147 or 1.25e7 < M < 1.3e38

   1. Initial program 86.7%

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

      1. times-frac 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in M around 0 76.7%

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

   if 3.8999999999999998e-147 < M < 1.25e7 or 1.3e38 < M

   1. Initial program 76.3%

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

      1. times-frac 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in M around 0 43.7%

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

      1. associate-*r/ 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*r/ 43.7%

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

      4. *-commutative 43.7%

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

      5. times-frac 45.0%

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

      6. unpow2 45.0%

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

      7. *-commutative 45.0%

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

      8. unpow2 45.0%

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

      9. unpow2 45.0%

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

   6. Simplified 45.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.9 \cdot 10^{-147}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 12500000 \lor \neg \left(M \leq 1.3 \cdot 10^{+38}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 75.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \end{array} \end{array} \]

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 4.8e-151)
   w0
   (* w0 (+ 1.0 (* -0.125 (/ (* D (* D (* M (* h M)))) (* d (* l d))))))))

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 <= 4.8e-151) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (M * (h * M)))) / (d * (l * d)))));
	}
	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 <= 4.8d-151) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d * (m * (h * m)))) / (d_1 * (l * d_1)))))
    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 <= 4.8e-151) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (M * (h * M)))) / (d * (l * d)))));
	}
	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 <= 4.8e-151:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (M * (h * M)))) / (d * (l * d)))))
	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 <= 4.8e-151)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D * Float64(M * Float64(h * M)))) / Float64(d * Float64(l * d))))));
	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 <= 4.8e-151)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D * (M * (h * M)))) / (d * (l * d)))));
	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, 4.8e-151], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.8e-151

   1. Initial program 86.8%

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

      1. times-frac 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in M around 0 76.3%

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

   if 4.8e-151 < M

   1. Initial program 77.0%

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

      1. times-frac 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in M around 0 44.7%

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

      1. associate-*r/ 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-*r/ 44.7%

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

      4. *-commutative 44.7%

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

      5. times-frac 46.0%

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

      6. unpow2 46.0%

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

      7. *-commutative 46.0%

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

      8. unpow2 46.0%

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

      9. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in D around 0 44.7%

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

      1. unpow2 44.7%

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

      2. associate-*r* 45.8%

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

      3. unpow2 45.8%

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

      4. associate-*l* 51.5%

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

      5. unpow2 51.5%

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

      6. associate-*r* 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \end{array} \]

Alternative 10: 72.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 (<= M 1.4e+80)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 (* M (* h M))) 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 (M <= 1.4e+80) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (h * M))) / 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 (m <= 1.4d+80) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * (m * (h * m))) / 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 (M <= 1.4e+80) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (h * M))) / 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 M <= 1.4e+80:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (h * M))) / 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 (M <= 1.4e+80)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * Float64(M * Float64(h * M))) / 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 (M <= 1.4e+80)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (h * M))) / 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[M, 1.4e+80], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.39999999999999992e80

   1. Initial program 84.6%

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

      1. times-frac 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in M around 0 76.2%

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

   if 1.39999999999999992e80 < M

   1. Initial program 76.2%

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

      1. times-frac 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in M around 0 30.1%

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

      1. associate-*r/ 30.1%

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

      2. *-commutative 30.1%

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

      3. associate-*r/ 30.1%

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

      4. *-commutative 30.1%

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

      5. times-frac 30.4%

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

      6. unpow2 30.4%

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

      7. *-commutative 30.4%

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

      8. unpow2 30.4%

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

      9. unpow2 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in D around 0 30.1%

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

      1. unpow2 30.1%

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

      2. associate-*r* 32.5%

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

      3. unpow2 32.5%

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

      4. associate-*l* 40.3%

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

      5. unpow2 40.3%

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

      6. associate-*r* 40.4%

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

   9. Simplified 40.4%

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

   10. Taylor expanded in D around inf 25.3%

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

       1. times-frac 23.3%

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

       2. unpow2 23.3%

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

       3. unpow2 23.3%

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

       4. times-frac 35.8%

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

       5. unpow2 35.8%

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

       6. unpow2 35.8%

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

       7. associate-*r* 35.9%

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

   12. Simplified 35.9%

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

       1. pow2 35.9%

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

   14. Applied egg-rr 35.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 11: 81.4% accurate, 10.3× speedup

Math

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

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) (/ (* h M) d)))) l) -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) * ((h * M) / d)))) / l) * -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) * ((h * m) / d_1)))) / l) * (-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) * ((h * M) / d)))) / l) * -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) * ((h * M) / d)))) / l) * -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(Float64(D * Float64(D * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))) / l) * -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) * ((h * M) / d)))) / l) * -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[(N[(D * N[(D * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

   1. times-frac 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in M around 0 54.0%

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

   1. associate-*r/ 54.0%

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

   2. *-commutative 54.0%

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

   3. associate-*r/ 54.0%

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

   4. *-commutative 54.0%

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

   5. times-frac 55.2%

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

   6. unpow2 55.2%

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

   7. *-commutative 55.2%

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

   8. unpow2 55.2%

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

   9. unpow2 55.2%

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

6. Simplified 55.2%

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

   1. associate-*l/ 56.8%

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

   2. times-frac 64.8%

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

8. Applied egg-rr 64.8%

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

   1. associate-*l* 71.8%

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

10. Simplified 71.8%

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

11. Taylor expanded in M around 0 63.9%

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

    1. unpow2 63.9%

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

    2. unpow2 63.9%

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

    3. associate-*r* 66.6%

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

    4. times-frac 75.7%

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

13. Simplified 75.7%

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

14. Final simplification 75.7%

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

Alternative 12: 68.6% 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 83.2%

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

   1. times-frac 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in M around 0 69.2%

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

5. Final simplification 69.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023213 
(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))))))