Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 87.4%

Time: 15.7s

Alternatives: 13

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 500:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 500.0)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) h) l))))
   (*
    w0
    (pow (+ 1.0 (* -0.25 (/ D (/ (/ d M) (/ (* h (* M D)) (* d l)))))) 0.5))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 500.0) {
		tmp = w0 * sqrt((1.0 - ((pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 500.0d0) then
        tmp = w0 * sqrt((1.0d0 - ((((m * (0.5d0 * (d / d_1))) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * ((1.0d0 + ((-0.25d0) * (d / ((d_1 / m) / ((h * (m * d)) / (d_1 * l)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 500.0) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 500.0:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((M * (0.5 * (D / d))), 2.0) * h) / l)))
	else:
		tmp = w0 * math.pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 500.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * (Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(d / M) / Float64(Float64(h * Float64(M * D)) / Float64(d * l)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 500.0)
		tmp = w0 * sqrt((1.0 - ((((M * (0.5 * (D / d))) ^ 2.0) * h) / l)));
	else
		tmp = w0 * ((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))) ^ 0.5);
	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: 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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 500.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 + N[(-0.25 * N[(D / N[(N[(d / M), $MachinePrecision] / N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 500

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 84.0%

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

   3. Simplified 84.0%

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

      1. unpow2 84.0%

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

      2. associate-*l* 85.7%

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

      3. div-inv 85.7%

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

      4. associate-*l* 83.8%

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

      5. times-frac 83.8%

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

      6. *-un-lft-identity 83.8%

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

      7. *-commutative 83.8%

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

      8. clear-num 83.8%

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

      9. div-inv 84.3%

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

      10. div-inv 84.3%

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

      11. associate-*l* 85.0%

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

      12. times-frac 85.0%

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

      13. *-un-lft-identity 85.0%

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

      14. *-commutative 85.0%

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

      15. clear-num 85.0%

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

      16. div-inv 85.5%

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

      17. associate-*l* 84.3%

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

   5. Applied egg-rr 90.1%

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

   if 500 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. times-frac 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in M around 0 35.7%

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

      1. associate-*r/ 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-*r/ 35.7%

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

      4. associate-/l* 35.7%

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

      5. unpow2 35.7%

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

      6. *-commutative 35.7%

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

      7. unpow2 35.7%

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

      8. associate-*l* 37.7%

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

      9. *-commutative 37.7%

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

      10. unpow2 37.7%

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

      11. associate-*l* 41.8%

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

   6. Simplified 41.8%

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

      1. pow1/2 41.8%

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

   8. Applied egg-rr 52.4%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 52.4%

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

      2. associate-/l* 53.1%

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

      3. associate-/l* 57.1%

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

   10. Applied egg-rr 57.1%

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

   11. Taylor expanded in D around 0 54.7%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{1 \cdot \frac{\frac{d}{M}}{\color{blue}{\frac{D \cdot \left(h \cdot M\right)}{d \cdot \ell}}}}\right)}^{0.5} \]
   12. Step-by-step derivation

       1. *-commutative 54.7%

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

       2. associate-*r* 56.7%

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

       3. *-commutative 56.7%

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

   13. Simplified 56.7%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{1 \cdot \frac{\frac{d}{M}}{\color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\ell \cdot d}}}}\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 500:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \]

Alternative 2: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 500:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{\frac{d}{M \cdot D}}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 500.0)
   (* w0 (sqrt (+ 1.0 (* -0.25 (/ (/ D (/ d (* M D))) (* l (/ d (* M h))))))))
   (*
    w0
    (pow (+ 1.0 (* -0.25 (/ D (/ (/ d M) (/ (* h (* M D)) (* d l)))))) 0.5))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 500.0) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 500.0d0) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (d_1 / (m * d))) / (l * (d_1 / (m * h)))))))
    else
        tmp = w0 * ((1.0d0 + ((-0.25d0) * (d / ((d_1 / m) / ((h * (m * d)) / (d_1 * l)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 500.0) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * Math.pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 500.0:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))))
	else:
		tmp = w0 * math.pow((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 500.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(d / Float64(M * D))) / Float64(l * Float64(d / Float64(M * h))))))));
	else
		tmp = Float64(w0 * (Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(d / M) / Float64(Float64(h * Float64(M * D)) / Float64(d * l)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 500.0)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	else
		tmp = w0 * ((1.0 + (-0.25 * (D / ((d / M) / ((h * (M * D)) / (d * l)))))) ^ 0.5);
	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: 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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 500.0], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 + N[(-0.25 * N[(D / N[(N[(d / M), $MachinePrecision] / N[(N[(h * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 500

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in M around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-*r/ 60.7%

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

      4. associate-/l* 61.2%

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

      5. unpow2 61.2%

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

      6. *-commutative 61.2%

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

      7. unpow2 61.2%

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

      8. associate-*l* 65.1%

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

      9. *-commutative 65.1%

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

      10. unpow2 65.1%

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

      11. associate-*l* 68.9%

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

   6. Simplified 68.9%

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

      1. pow1/2 68.9%

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

   8. Applied egg-rr 87.1%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 87.1%

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

      2. associate-/l* 88.5%

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

      3. associate-/l* 87.1%

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

   10. Applied egg-rr 87.1%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{1 \cdot \frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 87.1%

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

       2. unpow1/2 87.1%

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

       3. *-un-lft-identity 87.1%

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

       4. associate-/r/ 87.0%

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

       5. associate-/r/ 89.0%

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

       6. associate-/l/ 85.7%

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

   12. Applied egg-rr 85.7%

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

       1. *-lft-identity 85.7%

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

       2. associate-/r* 84.1%

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

       3. associate-/l/ 84.1%

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

       4. *-commutative 84.1%

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

       5. associate-/l/ 87.4%

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

   14. Simplified 87.4%

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

   if 500 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. times-frac 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in M around 0 35.7%

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

      1. associate-*r/ 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-*r/ 35.7%

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

      4. associate-/l* 35.7%

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

      5. unpow2 35.7%

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

      6. *-commutative 35.7%

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

      7. unpow2 35.7%

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

      8. associate-*l* 37.7%

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

      9. *-commutative 37.7%

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

      10. unpow2 37.7%

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

      11. associate-*l* 41.8%

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

   6. Simplified 41.8%

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

      1. pow1/2 41.8%

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

   8. Applied egg-rr 52.4%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 52.4%

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

      2. associate-/l* 53.1%

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

      3. associate-/l* 57.1%

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

   10. Applied egg-rr 57.1%

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

   11. Taylor expanded in D around 0 54.7%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{1 \cdot \frac{\frac{d}{M}}{\color{blue}{\frac{D \cdot \left(h \cdot M\right)}{d \cdot \ell}}}}\right)}^{0.5} \]
   12. Step-by-step derivation

       1. *-commutative 54.7%

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

       2. associate-*r* 56.7%

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

       3. *-commutative 56.7%

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

   13. Simplified 56.7%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{1 \cdot \frac{\frac{d}{M}}{\color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\ell \cdot d}}}}\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 500:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{\frac{d}{M \cdot D}}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \]

Alternative 3: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot w0}{\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}} \cdot \frac{D}{\ell}, w0\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{M} \cdot \left(\frac{d}{M} \cdot \frac{\ell}{h}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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) -4e+292)
   (fma -0.125 (* (/ (* D w0) (* (/ d M) (/ (/ d M) h))) (/ D l)) w0)
   (if (<= (/ h l) -2e-296)
     (*
      w0
      (sqrt (+ 1.0 (* -0.25 (* D (/ D (* (/ d M) (* (/ d M) (/ l h)))))))))
     (* w0 (+ 1.0 (* -0.125 (/ D (/ (* (/ d M) (/ (* d l) (* M h))) D))))))))

C

M = abs(M);
D = abs(D);
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) <= -4e+292) {
		tmp = fma(-0.125, (((D * w0) / ((d / M) * ((d / M) / h))) * (D / l)), w0);
	} else if ((h / l) <= -2e-296) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * ((d / M) * (l / h))))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D / (((d / M) * ((d * l) / (M * h))) / D))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -4e+292)
		tmp = fma(-0.125, Float64(Float64(Float64(D * w0) / Float64(Float64(d / M) * Float64(Float64(d / M) / h))) * Float64(D / l)), w0);
	elseif (Float64(h / l) <= -2e-296)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D * Float64(D / Float64(Float64(d / M) * Float64(Float64(d / M) * Float64(l / h)))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d * l) / Float64(M * h))) / D)))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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], -4e+292], N[(-0.125 * N[(N[(N[(D * w0), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-296], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D * N[(D / N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d * l), $MachinePrecision] / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -4.0000000000000001e292

   1. Initial program 43.3%

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

      1. *-commutative 43.3%

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

      2. times-frac 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in M around 0 53.1%

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

      1. fma-def 53.1%

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

      2. times-frac 53.1%

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

      3. *-commutative 53.1%

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

      4. unpow2 53.1%

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

      5. associate-/l* 58.8%

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

      6. *-commutative 58.8%

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

      7. *-commutative 58.8%

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

      8. unpow2 58.8%

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

      9. times-frac 58.8%

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

      10. *-commutative 58.8%

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

      11. unpow2 58.8%

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

      12. associate-*l* 58.8%

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

   6. Simplified 58.8%

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

   7. Taylor expanded in D around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. unpow2 53.1%

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

      3. associate-*r* 53.1%

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

      4. *-commutative 53.1%

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

      5. times-frac 53.1%

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

      6. unpow2 53.1%

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

      7. unpow2 53.1%

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

      8. associate-*l/ 58.8%

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

      9. *-commutative 58.8%

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

      10. associate-*r* 58.8%

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

   9. Simplified 74.8%

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

   if -4.0000000000000001e292 < (/.f64 h l) < -2e-296

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in M around 0 52.3%

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

      1. associate-*r/ 52.3%

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

      2. *-commutative 52.3%

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

      3. associate-*r/ 52.3%

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

      4. associate-/l* 53.1%

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

      5. unpow2 53.1%

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

      6. *-commutative 53.1%

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

      7. unpow2 53.1%

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

      8. associate-*l* 57.3%

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

      9. *-commutative 57.3%

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

      10. unpow2 57.3%

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

      11. associate-*l* 60.7%

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

   6. Simplified 60.7%

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

      1. *-un-lft-identity 60.7%

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

      2. cancel-sign-sub-inv 60.7%

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

      3. metadata-eval 60.7%

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

      4. associate-/l* 64.8%

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

      5. associate-*r* 60.0%

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

      6. *-commutative 60.0%

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

      7. associate-*r* 57.2%

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

      8. *-commutative 57.2%

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

      9. associate-*r* 61.5%

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

      10. associate-*r* 64.8%

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

      11. times-frac 75.8%

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

   8. Applied egg-rr 75.8%

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

   10. Simplified 81.3%

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

   if -2e-296 < (/.f64 h l)

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in M around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*r/ 61.9%

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

      4. *-commutative 61.9%

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

      5. associate-/l* 61.9%

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

      6. unpow2 61.9%

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

      7. unpow2 61.9%

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

      8. *-commutative 61.9%

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

      9. unpow2 61.9%

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

   6. Simplified 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. associate-/l* 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-*r* 77.3%

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

      5. associate-*r* 82.4%

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

      6. times-frac 89.8%

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

   8. Applied egg-rr 89.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot w0}{\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}} \cdot \frac{D}{\ell}, w0\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{M} \cdot \left(\frac{d}{M} \cdot \frac{\ell}{h}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)\\ \end{array} \]

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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) -5e-195)
   (*
    w0
    (pow (+ 1.0 (* -0.25 (/ D (/ (/ d M) (/ D (/ d (/ (* M h) l))))))) 0.5))
   (*
    w0
    (pow (+ 1.0 (* -0.25 (/ D (/ (/ d M) (/ (* D (* M h)) (* d l)))))) 0.5))))

C

M = abs(M);
D = abs(D);
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) <= -5e-195) {
		tmp = w0 * pow((1.0 + (-0.25 * (D / ((d / M) / (D / (d / ((M * h) / l))))))), 0.5);
	} else {
		tmp = w0 * pow((1.0 + (-0.25 * (D / ((d / M) / ((D * (M * h)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) <= (-5d-195)) then
        tmp = w0 * ((1.0d0 + ((-0.25d0) * (d / ((d_1 / m) / (d / (d_1 / ((m * h) / l))))))) ** 0.5d0)
    else
        tmp = w0 * ((1.0d0 + ((-0.25d0) * (d / ((d_1 / m) / ((d * (m * h)) / (d_1 * l)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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) <= -5e-195) {
		tmp = w0 * Math.pow((1.0 + (-0.25 * (D / ((d / M) / (D / (d / ((M * h) / l))))))), 0.5);
	} else {
		tmp = w0 * Math.pow((1.0 + (-0.25 * (D / ((d / M) / ((D * (M * h)) / (d * l)))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e-195:
		tmp = w0 * math.pow((1.0 + (-0.25 * (D / ((d / M) / (D / (d / ((M * h) / l))))))), 0.5)
	else:
		tmp = w0 * math.pow((1.0 + (-0.25 * (D / ((d / M) / ((D * (M * h)) / (d * l)))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-195)
		tmp = Float64(w0 * (Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(d / M) / Float64(D / Float64(d / Float64(Float64(M * h) / l))))))) ^ 0.5));
	else
		tmp = Float64(w0 * (Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(d / M) / Float64(Float64(D * Float64(M * h)) / Float64(d * l)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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) <= -5e-195)
		tmp = w0 * ((1.0 + (-0.25 * (D / ((d / M) / (D / (d / ((M * h) / l))))))) ^ 0.5);
	else
		tmp = w0 * ((1.0 + (-0.25 * (D / ((d / M) / ((D * (M * h)) / (d * l)))))) ^ 0.5);
	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: 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], -5e-195], N[(w0 * N[Power[N[(1.0 + N[(-0.25 * N[(D / N[(N[(d / M), $MachinePrecision] / N[(D / N[(d / N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 + N[(-0.25 * N[(D / N[(N[(d / M), $MachinePrecision] / N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.00000000000000009e-195

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. times-frac 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in M around 0 52.1%

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

      1. associate-*r/ 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*r/ 52.1%

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

      4. associate-/l* 52.9%

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

      5. unpow2 52.9%

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

      6. *-commutative 52.9%

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

      7. unpow2 52.9%

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

      8. associate-*l* 55.9%

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

      9. *-commutative 55.9%

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

      10. unpow2 55.9%

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

      11. associate-*l* 58.8%

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

   6. Simplified 58.8%

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

      1. pow1/2 58.8%

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

   8. Applied egg-rr 72.0%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 72.0%

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

      2. associate-/l* 74.4%

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

      3. associate-/l* 76.6%

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

   10. Applied egg-rr 76.6%

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

   if -5.00000000000000009e-195 < (/.f64 h l)

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. times-frac 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 60.1%

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

      1. associate-*r/ 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*r/ 60.1%

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

      4. associate-/l* 60.1%

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

      5. unpow2 60.1%

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

      6. *-commutative 60.1%

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

      7. unpow2 60.1%

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

      8. associate-*l* 64.2%

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

      9. *-commutative 64.2%

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

      10. unpow2 64.2%

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

      11. associate-*l* 69.1%

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

   6. Simplified 69.1%

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

      1. pow1/2 69.1%

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

   8. Applied egg-rr 89.8%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 86.6%

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

   10. Applied egg-rr 86.6%

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

   11. Taylor expanded in M around -inf 91.3%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{1 \cdot \frac{\frac{d}{M}}{\color{blue}{\frac{D \cdot \left(h \cdot M\right)}{\ell \cdot d}}}}\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{\frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}}}\right)}^{0.5}\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{M \cdot D}\\ \mathbf{if}\;D \leq 4 \cdot 10^{+137}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{t_0}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{t_0 \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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
 (let* ((t_0 (/ d (* M D))))
   (if (<= D 4e+137)
     (* w0 (sqrt (+ 1.0 (* -0.25 (/ (/ D t_0) (* l (/ d (* M h))))))))
     (* w0 (pow (+ 1.0 (* -0.25 (/ D (* t_0 (* l (/ (/ d h) M)))))) 0.5)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d / (M * D);
	double tmp;
	if (D <= 4e+137) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / t_0) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * pow((1.0 + (-0.25 * (D / (t_0 * (l * ((d / h) / M)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) :: t_0
    real(8) :: tmp
    t_0 = d_1 / (m * d)
    if (d <= 4d+137) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / t_0) / (l * (d_1 / (m * h)))))))
    else
        tmp = w0 * ((1.0d0 + ((-0.25d0) * (d / (t_0 * (l * ((d_1 / h) / m)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d / (M * D);
	double tmp;
	if (D <= 4e+137) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / t_0) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * Math.pow((1.0 + (-0.25 * (D / (t_0 * (l * ((d / h) / M)))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = d / (M * D)
	tmp = 0
	if D <= 4e+137:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / t_0) / (l * (d / (M * h)))))))
	else:
		tmp = w0 * math.pow((1.0 + (-0.25 * (D / (t_0 * (l * ((d / h) / M)))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(d / Float64(M * D))
	tmp = 0.0
	if (D <= 4e+137)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / t_0) / Float64(l * Float64(d / Float64(M * h))))))));
	else
		tmp = Float64(w0 * (Float64(1.0 + Float64(-0.25 * Float64(D / Float64(t_0 * Float64(l * Float64(Float64(d / h) / M)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = d / (M * D);
	tmp = 0.0;
	if (D <= 4e+137)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / t_0) / (l * (d / (M * h)))))));
	else
		tmp = w0 * ((1.0 + (-0.25 * (D / (t_0 * (l * ((d / h) / M)))))) ^ 0.5);
	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: 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_] := Block[{t$95$0 = N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 4e+137], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / t$95$0), $MachinePrecision] / N[(l * N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 + N[(-0.25 * N[(D / N[(t$95$0 * N[(l * N[(N[(d / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 4.0000000000000001e137

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 80.2%

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

   3. Simplified 80.2%

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

   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. associate-*r/ 58.9%

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

      4. associate-/l* 59.4%

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

      5. unpow2 59.4%

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

      6. *-commutative 59.4%

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

      7. unpow2 59.4%

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

      8. associate-*l* 63.1%

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

      9. *-commutative 63.1%

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

      10. unpow2 63.1%

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

      11. associate-*l* 66.8%

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

   6. Simplified 66.8%

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

      1. pow1/2 66.8%

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

   8. Applied egg-rr 82.1%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 82.1%

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

      2. associate-/l* 83.2%

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

      3. associate-/l* 82.8%

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

   10. Applied egg-rr 82.8%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{1 \cdot \frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 82.8%

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

       2. unpow1/2 82.7%

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

       3. *-un-lft-identity 82.7%

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

       4. associate-/r/ 82.6%

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

       5. associate-/r/ 84.5%

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

       6. associate-/l/ 80.9%

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

   12. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

       2. associate-/r* 79.0%

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

       3. associate-/l/ 79.6%

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

       4. *-commutative 79.6%

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

       5. associate-/l/ 83.1%

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

   14. Simplified 83.1%

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

   if 4.0000000000000001e137 < D

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. times-frac 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in M around 0 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-*r/ 39.0%

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

      4. associate-/l* 39.0%

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

      5. unpow2 39.0%

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

      6. *-commutative 39.0%

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

      7. unpow2 39.0%

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

      8. associate-*l* 41.6%

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

      9. *-commutative 41.6%

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

      10. unpow2 41.6%

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

      11. associate-*l* 46.5%

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

   6. Simplified 46.5%

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

      1. pow1/2 46.5%

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

   8. Applied egg-rr 71.5%

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

   9. Taylor expanded in d around 0 58.2%

      \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{\frac{{d}^{2} \cdot \ell}{D \cdot \left({M}^{2} \cdot h\right)}}}\right)}^{0.5} \]
   10. Step-by-step derivation

       1. *-commutative 58.2%

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

       2. unpow2 58.2%

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

       3. associate-*r* 65.5%

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

       4. associate-/r* 63.1%

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

       5. associate-*r* 58.2%

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

       6. times-frac 53.3%

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

       7. associate-/r* 64.0%

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

       8. associate-/r/ 66.4%

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

       9. unpow2 66.4%

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

       10. associate-*r/ 69.0%

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

       11. *-commutative 69.0%

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

       12. associate-/l* 68.8%

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

       13. associate-*r/ 69.1%

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

       14. associate-*l/ 73.4%

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

       15. *-commutative 73.4%

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

       16. associate-/r/ 76.2%

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

       17. associate-/r* 78.5%

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

       18. *-commutative 78.5%

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

       19. associate-/l/ 76.2%

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

       20. associate-/r* 78.6%

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

       21. associate-/r* 76.0%

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

   11. Simplified 76.0%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{\left(\ell \cdot \frac{\frac{d}{h}}{M}\right) \cdot \frac{d}{M \cdot D}}}\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4 \cdot 10^{+137}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{\frac{d}{M \cdot D}}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\frac{d}{M \cdot D} \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}\right)}^{0.5}\\ \end{array} \]

Alternative 6: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 10^{+167}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{\frac{M \cdot h}{\ell}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 1e+167)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* D (/ D (* (/ d M) (/ d (/ (* M h) l)))))))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (/ D (* (/ (/ d M) D) (* l (/ (/ d h) M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1e+167) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 1d+167) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (d * (d / ((d_1 / m) * (d_1 / ((m * h) / l))))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (d / (((d_1 / m) / d) * (l * ((d_1 / h) / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 1e+167) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1e+167:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1e+167)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D * Float64(D / Float64(Float64(d / M) * Float64(d / Float64(Float64(M * h) / l)))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(Float64(d / M) / D) * Float64(l * Float64(Float64(d / h) / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1e+167)
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	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: 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, 1e+167], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D * N[(D / N[(N[(d / M), $MachinePrecision] * N[(d / N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision] * N[(l * N[(N[(d / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1e167

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 58.5%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. associate-*r/ 58.5%

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

      4. associate-/l* 59.0%

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

      5. unpow2 59.0%

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

      6. *-commutative 59.0%

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

      7. unpow2 59.0%

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

      8. associate-*l* 62.5%

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

      9. *-commutative 62.5%

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

      10. unpow2 62.5%

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

      11. associate-*l* 66.5%

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

   6. Simplified 66.5%

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

      1. pow1/2 66.5%

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

   8. Applied egg-rr 81.8%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 81.8%

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

      2. associate-/r/ 81.8%

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

      3. *-commutative 81.8%

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

      4. associate-/l* 81.4%

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

   10. Applied egg-rr 81.4%

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

   if 1e167 < D

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. times-frac 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in M around 0 37.9%

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

      1. associate-*r/ 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-*r/ 37.9%

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

      4. associate-/l* 37.9%

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

      5. unpow2 37.9%

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

      6. *-commutative 37.9%

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

      7. unpow2 37.9%

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

      8. associate-*l* 41.0%

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

      9. *-commutative 41.0%

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

      10. unpow2 41.0%

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

      11. associate-*l* 44.2%

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

   6. Simplified 44.2%

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

      1. pow1/2 44.2%

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

   8. Applied egg-rr 71.2%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 71.2%

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

      2. associate-/l* 74.0%

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

      3. associate-/l* 73.9%

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

   10. Applied egg-rr 73.9%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{1 \cdot \frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. unpow1/2 73.9%

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

       2. *-un-lft-identity 73.9%

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

       3. associate-/r/ 73.5%

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

       4. associate-/r/ 77.0%

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

       5. associate-/l/ 79.8%

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

   12. Applied egg-rr 79.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 10^{+167}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{\frac{M \cdot h}{\ell}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}}\\ \end{array} \]

Alternative 7: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4.4 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{\frac{d}{M \cdot D}}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 4.4e+139)
   (* w0 (sqrt (+ 1.0 (* -0.25 (/ (/ D (/ d (* M D))) (* l (/ d (* M h))))))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (/ D (* (/ (/ d M) D) (* l (/ (/ d h) M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.4e+139) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 4.4d+139) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (d_1 / (m * d))) / (l * (d_1 / (m * h)))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (d / (((d_1 / m) / d) * (l * ((d_1 / h) / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 4.4e+139) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 4.4e+139:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.4e+139)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(d / Float64(M * D))) / Float64(l * Float64(d / Float64(M * h))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D / Float64(Float64(Float64(d / M) / D) * Float64(l * Float64(Float64(d / h) / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 4.4e+139)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (d / (M * D))) / (l * (d / (M * h)))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (D / (((d / M) / D) * (l * ((d / h) / M)))))));
	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: 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, 4.4e+139], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision] * N[(l * N[(N[(d / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.3999999999999999e139

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 80.2%

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

   3. Simplified 80.2%

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

   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. associate-*r/ 58.9%

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

      4. associate-/l* 59.4%

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

      5. unpow2 59.4%

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

      6. *-commutative 59.4%

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

      7. unpow2 59.4%

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

      8. associate-*l* 63.1%

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

      9. *-commutative 63.1%

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

      10. unpow2 63.1%

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

      11. associate-*l* 66.8%

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

   6. Simplified 66.8%

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

      1. pow1/2 66.8%

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

   8. Applied egg-rr 82.1%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 82.1%

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

      2. associate-/l* 83.2%

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

      3. associate-/l* 82.8%

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

   10. Applied egg-rr 82.8%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{1 \cdot \frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 82.8%

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

       2. unpow1/2 82.7%

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

       3. *-un-lft-identity 82.7%

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

       4. associate-/r/ 82.6%

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

       5. associate-/r/ 84.5%

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

       6. associate-/l/ 80.9%

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

   12. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

       2. associate-/r* 79.0%

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

       3. associate-/l/ 79.6%

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

       4. *-commutative 79.6%

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

       5. associate-/l/ 83.1%

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

   14. Simplified 83.1%

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

   if 4.3999999999999999e139 < D

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. times-frac 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in M around 0 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-*r/ 39.0%

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

      4. associate-/l* 39.0%

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

      5. unpow2 39.0%

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

      6. *-commutative 39.0%

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

      7. unpow2 39.0%

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

      8. associate-*l* 41.6%

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

      9. *-commutative 41.6%

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

      10. unpow2 41.6%

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

      11. associate-*l* 46.5%

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

   6. Simplified 46.5%

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

      1. pow1/2 46.5%

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

   8. Applied egg-rr 71.5%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 71.5%

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

      2. associate-/l* 73.8%

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

      3. associate-/l* 73.7%

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

   10. Applied egg-rr 73.7%

       \[\leadsto w0 \cdot {\left(1 + -0.25 \cdot \frac{D}{\color{blue}{1 \cdot \frac{\frac{d}{M}}{\frac{D}{\frac{d}{\frac{M \cdot h}{\ell}}}}}}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. unpow1/2 73.7%

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

       2. *-un-lft-identity 73.7%

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

       3. associate-/r/ 73.4%

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

       4. associate-/r/ 76.2%

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

       5. associate-/l/ 78.6%

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

   12. Applied egg-rr 78.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.4 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\frac{D}{\frac{d}{M \cdot D}}}{\ell \cdot \frac{d}{M \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M}}{D} \cdot \left(\ell \cdot \frac{\frac{d}{h}}{M}\right)}}\\ \end{array} \]

Alternative 8: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{\frac{M \cdot h}{\ell}}}\right)} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 (sqrt (+ 1.0 (* -0.25 (* D (/ D (* (/ d M) (/ d (/ (* M h) l))))))))))

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 * sqrt((1.0d0 + ((-0.25d0) * (d * (d / ((d_1 / m) * (d_1 / ((m * h) / l))))))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 * Math.sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))));
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))))

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D * Float64(D / Float64(Float64(d / M) * Float64(d / Float64(Float64(M * h) / l)))))))))
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-0.25 * (D * (D / ((d / M) * (d / ((M * h) / l))))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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[Sqrt[N[(1.0 + N[(-0.25 * N[(D * N[(D / N[(N[(d / M), $MachinePrecision] * N[(d / N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. *-commutative 80.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in M around 0 55.9%

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

   1. associate-*r/ 55.9%

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

   2. *-commutative 55.9%

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

   3. associate-*r/ 55.9%

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

   4. associate-/l* 56.3%

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

   5. unpow2 56.3%

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

   6. *-commutative 56.3%

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

   7. unpow2 56.3%

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

   8. associate-*l* 59.8%

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

   9. *-commutative 59.8%

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

   10. unpow2 59.8%

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

   11. associate-*l* 63.7%

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

6. Simplified 63.7%

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

   1. pow1/2 63.7%

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

8. Applied egg-rr 80.5%

   \[\leadsto w0 \cdot \color{blue}{{\left(1 + -0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d \cdot \ell}{M \cdot h}}{D}}\right)}^{0.5}} \]
9. Step-by-step derivation

   1. unpow1/2 80.5%

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

   2. associate-/r/ 80.5%

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

   3. *-commutative 80.5%

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

   4. associate-/l* 79.7%

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

10. Applied egg-rr 79.7%

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

11. Final simplification 79.7%

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

Alternative 9: 74.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-168}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 1.7e-168)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (* (/ M d) (/ (* M h) d))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.7e-168) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 1.7d-168) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((m / d_1) * ((m * h) / d_1)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 1.7e-168) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.7e-168:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.7e-168)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M / d) * Float64(Float64(M * h) / d))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.7e-168)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / 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: 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, 1.7e-168], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.70000000000000011e-168

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. times-frac 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in M around 0 65.6%

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

   if 1.70000000000000011e-168 < M

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in M around 0 60.0%

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

      1. associate-*r/ 60.0%

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

      2. *-commutative 60.0%

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

      3. associate-*r/ 60.0%

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

      4. *-commutative 60.0%

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

      5. associate-/l* 60.1%

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

      6. unpow2 60.1%

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

      7. unpow2 60.1%

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

      8. *-commutative 60.1%

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

      9. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in D around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. times-frac 60.1%

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

      3. unpow2 60.1%

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

      4. associate-*r/ 60.2%

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

      5. unpow2 60.2%

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

      6. unpow2 60.2%

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

      7. associate-*r* 61.7%

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

      8. times-frac 65.7%

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

   9. Simplified 65.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-168}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \end{array} \]

Alternative 10: 77.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 8.6 \cdot 10^{-180}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 8.6e-180)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D (* d l))) (* M (* M h))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 8.6e-180) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / (d * l))) * (M * (M * h)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 8.6d-180) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / (d_1 * l))) * (m * (m * h)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 8.6e-180) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / (d * l))) * (M * (M * h)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 8.6e-180:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / (d * l))) * (M * (M * h)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 8.6e-180)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / Float64(d * l))) * Float64(M * Float64(M * h))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 8.6e-180)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / (d * l))) * (M * (M * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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, 8.6e-180], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.5999999999999991e-180

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. times-frac 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in M around 0 65.2%

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

   if 8.5999999999999991e-180 < M

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 61.0%

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

      1. associate-*r/ 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*r/ 61.0%

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

      4. *-commutative 61.0%

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

      5. associate-/l* 61.1%

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

      6. unpow2 61.1%

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

      7. unpow2 61.1%

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

      8. *-commutative 61.1%

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

      9. unpow2 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in D around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-*r* 63.6%

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

      3. unpow2 63.6%

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

      4. associate-*r* 66.1%

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

      5. associate-*l/ 65.1%

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

      6. unpow2 65.1%

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

      7. times-frac 66.6%

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

      8. *-commutative 66.6%

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

   9. Simplified 66.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.6 \cdot 10^{-180}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 72.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 23000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{w0 \cdot h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 23000000000.0)
   w0
   (* -0.125 (* (* D (/ D l)) (* (/ (* w0 h) d) (/ (* M M) d))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 23000000000.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((w0 * h) / d) * ((M * M) / d)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 23000000000.0d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * (((w0 * h) / d_1) * ((m * m) / d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 23000000000.0) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((w0 * h) / d) * ((M * M) / d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 23000000000.0:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * (((w0 * h) / d) * ((M * M) / d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 23000000000.0)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(w0 * h) / d) * Float64(Float64(M * M) / d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 23000000000.0)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * (((w0 * h) / d) * ((M * M) / 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: 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, 23000000000.0], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w0 * h), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.3e10

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in M around 0 63.9%

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

   if 2.3e10 < 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. *-commutative 76.2%

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

      2. times-frac 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 51.8%

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

      1. associate-*r/ 51.8%

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

      2. *-commutative 51.8%

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

      3. associate-*r/ 51.8%

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

      4. *-commutative 51.8%

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

      5. associate-/l* 51.8%

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

      6. unpow2 51.8%

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

      7. unpow2 51.8%

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

      8. *-commutative 51.8%

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

      9. unpow2 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in D around 0 51.8%

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

      1. *-commutative 51.8%

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

      2. times-frac 51.8%

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

      3. unpow2 51.8%

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

      4. associate-*r/ 52.0%

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

      5. unpow2 52.0%

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

      6. unpow2 52.0%

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

      7. associate-*r* 54.6%

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

      8. times-frac 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in D around inf 21.9%

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

       1. associate-*r/ 21.9%

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

       2. *-commutative 21.9%

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

       3. unpow2 21.9%

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

       4. *-commutative 21.9%

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

       5. unpow2 21.9%

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

       6. associate-*r/ 21.9%

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

       7. times-frac 24.0%

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

       8. unpow2 24.0%

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

       9. associate-*r/ 24.2%

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

       10. unpow2 24.2%

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

       11. unpow2 24.2%

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

       12. associate-*r* 24.3%

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

   12. Simplified 24.3%

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

       1. times-frac 26.8%

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

       2. *-commutative 26.8%

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

   14. Applied egg-rr 26.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 23000000000:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{w0 \cdot h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\\ \end{array} \]

Alternative 12: 78.4% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. *-commutative 80.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in M around 0 54.5%

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

   1. associate-*r/ 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*r/ 54.5%

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

   4. *-commutative 54.5%

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

   5. associate-/l* 54.9%

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

   6. unpow2 54.9%

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

   7. unpow2 54.9%

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

   8. *-commutative 54.9%

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

   9. unpow2 54.9%

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

6. Simplified 54.9%

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

   1. *-un-lft-identity 54.9%

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

   2. associate-/l* 62.4%

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

   3. *-commutative 62.4%

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

   4. associate-*r* 65.3%

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

   5. associate-*r* 68.3%

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

   6. times-frac 74.5%

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

8. Applied egg-rr 74.5%

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

9. Final simplification 74.5%

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

Alternative 13: 67.8% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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);
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: 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);
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)
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)
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)
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: 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 80.2%

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

   1. *-commutative 80.2%

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

   2. times-frac 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in M around 0 61.0%

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

5. Final simplification 61.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023228 
(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))))))