Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 85.6%

Time: 15.3s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-246}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \end{array} \]

FPCore

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) -1e-246)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (* (/ d M) (/ (/ d M) h)) (/ l D)))))))))

C

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) <= -1e-246) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	}
	return tmp;
}

Fortran

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) <= (-1d-246)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 / m) * ((d_1 / m) / h)) * (l / d))))))
    end if
    code = tmp
end function

Java

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) <= -1e-246) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -1e-246:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-246)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d / M) / h)) * Float64(l / D)))))));
	end
	return tmp
end

MATLAB

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) <= -1e-246)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	end
	tmp_2 = tmp;
end

Wolfram

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], -1e-246], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.99999999999999956e-247

   1. Initial program 79.7%

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

      1. times-frac 79.8%

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

   3. Simplified 79.8%

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

      1. associate-*r/ 84.1%

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

      2. frac-times 84.0%

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

      3. div-inv 84.1%

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

      4. associate-*l* 84.1%

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

      5. associate-/r* 84.1%

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

      6. metadata-eval 84.1%

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

   5. Applied egg-rr 84.1%

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

   if -9.99999999999999956e-247 < (/.f64 h l)

   1. Initial program 84.1%

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

      1. times-frac 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in M around 0 52.9%

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

      1. associate-*r/ 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-*r/ 52.9%

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

      4. associate-/l* 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. unpow2 52.9%

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

      7. *-commutative 52.9%

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

      8. unpow2 52.9%

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

   6. Simplified 52.9%

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

      1. expm1-log1p-u 50.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 50.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 61.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 61.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 61.9%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 61.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 64.8%

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

      3. associate-/l* 65.6%

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

      4. associate-/l* 72.3%

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

   10. Simplified 72.3%

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

       1. associate-/r/ 73.2%

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

   12. Applied egg-rr 73.2%

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

   13. Taylor expanded in l around 0 64.0%

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

       1. *-commutative 64.0%

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

       2. *-commutative 64.0%

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

       3. unpow2 64.0%

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

       4. *-commutative 64.0%

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

       5. associate-*r* 69.7%

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

       6. times-frac 72.3%

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

       7. unpow2 72.3%

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

       8. *-commutative 72.3%

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

       9. times-frac 90.0%

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

       10. associate-/r* 92.6%

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

   15. Simplified 92.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-246}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \]

Alternative 2: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}} \leq 10^{+200}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \end{array} \]

FPCore

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 (<=
      (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* M D) (* d 2.0)) 2.0)))))
      1e+200)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (* (/ d M) (/ (/ d M) h)) (/ l D)))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - ((h / l) * pow(((M * D) / (d * 2.0)), 2.0))))) <= 1e+200) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	}
	return tmp;
}

Fortran

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 ((w0 * sqrt((1.0d0 - ((h / l) * (((m * d) / (d_1 * 2.0d0)) ** 2.0d0))))) <= 1d+200) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / 2.0d0) * (d / d_1)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 / m) * ((d_1 / m) / h)) * (l / d))))))
    end if
    code = tmp
end function

Java

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 ((w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 1e+200) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 1e+200:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0))))) <= 1e+200)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d / M) / h)) * Float64(l / D)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((h / l) * (((M * D) / (d * 2.0)) ^ 2.0))))) <= 1e+200)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	end
	tmp_2 = tmp;
end

Wolfram

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[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+200], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 9.9999999999999997e199

   1. Initial program 90.3%

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

      1. times-frac 90.4%

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

   3. Simplified 90.4%

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

   if 9.9999999999999997e199 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 51.7%

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

      1. times-frac 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in M around 0 50.3%

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

      1. associate-*r/ 50.3%

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

      2. *-commutative 50.3%

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

      3. associate-*r/ 50.3%

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

      4. associate-/l* 50.3%

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

      5. unpow2 50.3%

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

      6. unpow2 50.3%

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

      7. *-commutative 50.3%

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

      8. unpow2 50.3%

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

   6. Simplified 50.3%

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

      1. expm1-log1p-u 21.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 21.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 21.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 21.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 21.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 21.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 51.8%

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

      3. associate-/l* 51.8%

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

      4. associate-/l* 57.1%

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

   10. Simplified 57.1%

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

       1. associate-/r/ 60.7%

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

   12. Applied egg-rr 60.7%

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

   13. Taylor expanded in l around 0 52.0%

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

       1. *-commutative 52.0%

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

       2. *-commutative 52.0%

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

       3. unpow2 52.0%

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

       4. *-commutative 52.0%

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

       5. associate-*r* 53.7%

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

       6. times-frac 57.0%

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

       7. unpow2 57.0%

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

       8. *-commutative 57.0%

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

       9. times-frac 78.1%

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

       10. associate-/r* 79.8%

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

   15. Simplified 79.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}} \leq 10^{+200}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \end{array} \]

Alternative 3: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.4 \cdot 10^{-47}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{d \cdot d}{D} \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h \cdot M}\right)}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 7.4e-47)
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (* (/ d M) (/ (/ d M) h)) (/ l D)))))))
   (if (<= d 1.15e+154)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (/ (* d d) D) (/ l (* M (* h M)))))))))
     (* w0 (+ 1.0 (* (* D (/ D (* l (* (/ d M) (/ d (* h M)))))) -0.125))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.4e-47) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else if (d <= 1.15e+154) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d * d) / D) * (l / (M * (h * M))))))));
	} else {
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 7.4d-47) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 / m) * ((d_1 / m) / h)) * (l / d))))))
    else if (d_1 <= 1.15d+154) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 * d_1) / d) * (l / (m * (h * m))))))))
    else
        tmp = w0 * (1.0d0 + ((d * (d / (l * ((d_1 / m) * (d_1 / (h * m)))))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 7.4e-47) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else if (d <= 1.15e+154) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d * d) / D) * (l / (M * (h * M))))))));
	} else {
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 7.4e-47:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))))
	elif d <= 1.15e+154:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d * d) / D) * (l / (M * (h * M))))))))
	else:
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 7.4e-47)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d / M) / h)) * Float64(l / D)))))));
	elseif (d <= 1.15e+154)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d * d) / D) * Float64(l / Float64(M * Float64(h * M)))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(D / Float64(l * Float64(Float64(d / M) * Float64(d / Float64(h * M)))))) * -0.125)));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 7.4e-47)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	elseif (d <= 1.15e+154)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d * d) / D) * (l / (M * (h * M))))))));
	else
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 7.4e-47], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e+154], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision] * N[(l / N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(D / N[(l * N[(N[(d / M), $MachinePrecision] * N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.4000000000000001e-47

   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. times-frac 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in M around 0 47.2%

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

      1. associate-*r/ 47.2%

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

      2. *-commutative 47.2%

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

      3. associate-*r/ 47.2%

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

      4. associate-/l* 46.1%

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

      5. unpow2 46.1%

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

      6. unpow2 46.1%

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

      7. *-commutative 46.1%

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

      8. unpow2 46.1%

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

   6. Simplified 46.1%

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

      1. expm1-log1p-u 31.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 31.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 37.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 35.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 35.6%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 35.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 51.4%

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

      3. associate-/l* 51.3%

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

      4. associate-/l* 58.5%

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

   10. Simplified 58.5%

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

       1. associate-/r/ 62.2%

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

   12. Applied egg-rr 62.2%

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

   13. Taylor expanded in l around 0 54.8%

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

       1. *-commutative 54.8%

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

       2. *-commutative 54.8%

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

       3. unpow2 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r* 57.7%

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

       6. times-frac 59.4%

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

       7. unpow2 59.4%

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

       8. *-commutative 59.4%

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

       9. times-frac 78.0%

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

       10. associate-/r* 79.2%

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

   15. Simplified 79.2%

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

   if 7.4000000000000001e-47 < d < 1.15e154

   1. Initial program 89.7%

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

      1. times-frac 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in M around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. *-commutative 66.5%

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

      3. associate-*r/ 66.5%

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

      4. associate-/l* 66.4%

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

      5. unpow2 66.4%

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

      6. unpow2 66.4%

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

      7. *-commutative 66.4%

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

      8. unpow2 66.4%

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

   6. Simplified 66.4%

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

      1. expm1-log1p-u 55.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 55.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 64.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 66.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 66.3%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 66.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 77.6%

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

      3. associate-/l* 79.4%

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

      4. associate-/l* 79.4%

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

   10. Simplified 79.4%

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

   11. Taylor expanded in l around 0 77.3%

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

       1. unpow2 77.3%

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

       2. *-commutative 77.3%

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

       3. times-frac 75.3%

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

       4. unpow2 75.3%

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

       5. *-commutative 75.3%

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

       6. associate-*r* 81.3%

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

       7. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if 1.15e154 < d

   1. Initial program 83.7%

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

      1. times-frac 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in M around 0 45.2%

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

      1. *-commutative 45.2%

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

      2. *-commutative 45.2%

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

      3. associate-/l* 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. *-commutative 47.6%

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

      7. unpow2 47.6%

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

   6. Simplified 47.6%

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

      1. expm1-log1p-u 47.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 47.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 57.4%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 59.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 59.8%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. expm1-def 59.8%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)} \cdot -0.125\right) \]

      2. expm1-log1p 59.8%

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

      3. associate-/r/ 59.8%

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

      4. unpow2 59.8%

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

      5. unpow2 59.8%

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

      6. times-frac 57.4%

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

      7. *-commutative 57.4%

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

      8. times-frac 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

   10. Simplified 57.4%

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

   11. Taylor expanded in l around 0 57.4%

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

       1. unpow2 57.4%

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

       2. unpow2 57.4%

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

       3. *-rgt-identity 57.4%

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

       4. associate-*r* 60.0%

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

       5. associate-*r/ 60.0%

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

       6. associate-*l* 60.0%

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

       7. associate-*r/ 60.0%

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

       8. *-rgt-identity 60.0%

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

       9. times-frac 77.2%

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

       10. *-commutative 77.2%

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

   13. Simplified 77.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.4 \cdot 10^{-47}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{d \cdot d}{D} \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h \cdot M}\right)}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 4: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h \cdot M}\right)}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

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 6.2e+139)
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (* (/ d M) (/ (/ d M) h)) (/ l D)))))))
   (* w0 (+ 1.0 (* (* D (/ D (* l (* (/ d M) (/ d (* h M)))))) -0.125)))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.2e+139) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else {
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 6.2d+139) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 / m) * ((d_1 / m) / h)) * (l / d))))))
    else
        tmp = w0 * (1.0d0 + ((d * (d / (l * ((d_1 / m) * (d_1 / (h * m)))))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

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 <= 6.2e+139) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else {
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6.2e+139:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))))
	else:
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6.2e+139)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d / M) / h)) * Float64(l / D)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(D / Float64(l * Float64(Float64(d / M) * Float64(d / Float64(h * M)))))) * -0.125)));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6.2e+139)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	else
		tmp = w0 * (1.0 + ((D * (D / (l * ((d / M) * (d / (h * M)))))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

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, 6.2e+139], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(D * N[(D / N[(l * N[(N[(d / M), $MachinePrecision] * N[(d / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.2e139

   1. Initial program 80.8%

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

      1. times-frac 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in M around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*r/ 51.5%

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

      4. associate-/l* 50.6%

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

      5. unpow2 50.6%

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

      6. unpow2 50.6%

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

      7. *-commutative 50.6%

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

      8. unpow2 50.6%

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

   6. Simplified 50.6%

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

      1. expm1-log1p-u 36.5%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 36.5%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 42.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 41.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 41.6%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 41.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 56.4%

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

      3. associate-/l* 56.7%

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

      4. associate-/l* 62.4%

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

   10. Simplified 62.4%

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

       1. associate-/r/ 65.5%

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

   12. Applied egg-rr 65.5%

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

   13. Taylor expanded in l around 0 58.6%

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

       1. *-commutative 58.6%

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

       2. *-commutative 58.6%

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

       3. unpow2 58.6%

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

       4. *-commutative 58.6%

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

       5. associate-*r* 61.9%

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

       6. times-frac 63.2%

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

       7. unpow2 63.2%

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

       8. *-commutative 63.2%

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

       9. times-frac 79.1%

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

       10. associate-/r* 80.0%

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

   15. Simplified 80.0%

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

   if 6.2e139 < d

   1. Initial program 85.7%

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

      1. times-frac 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in M around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-/l* 47.9%

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

      4. unpow2 47.9%

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

      5. unpow2 47.9%

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

      6. *-commutative 47.9%

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

      7. unpow2 47.9%

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

   6. Simplified 47.9%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 45.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 58.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 60.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 60.8%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. expm1-def 60.8%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)} \cdot -0.125\right) \]

      2. expm1-log1p 62.9%

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

      3. associate-/r/ 62.9%

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

      4. unpow2 62.9%

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

      5. unpow2 62.9%

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

      6. times-frac 60.8%

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

      7. *-commutative 60.8%

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

      8. times-frac 60.8%

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

      9. unpow2 60.8%

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

      10. unpow2 60.8%

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

   10. Simplified 60.8%

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

   11. Taylor expanded in l around 0 60.8%

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

       1. unpow2 60.8%

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

       2. unpow2 60.8%

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

       3. *-rgt-identity 60.8%

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

       4. associate-*r* 63.1%

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

       5. associate-*r/ 63.1%

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

       6. associate-*l* 63.1%

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

       7. associate-*r/ 63.1%

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

       8. *-rgt-identity 63.1%

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

       9. times-frac 78.1%

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

       10. *-commutative 78.1%

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

   13. Simplified 78.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h \cdot M}\right)}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\ell}{M \cdot M}}{\frac{D}{d} \cdot \frac{h}{d}}}}\\ \end{array} \end{array} \]

FPCore

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-24)
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (* (* (/ d M) (/ (/ d M) h)) (/ l D)))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (/ D (/ (/ l (* M M)) (* (/ D d) (/ h d))))))))))

C

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-24) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((l / (M * M)) / ((D / d) * (h / d)))))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1d-24) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / (((d_1 / m) * ((d_1 / m) / h)) * (l / d))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d / ((l / (m * m)) / ((d / d_1) * (h / d_1)))))))
    end if
    code = tmp
end function

Java

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-24) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D / ((l / (M * M)) / ((D / d) * (h / d)))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1e-24:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D / ((l / (M * M)) / ((D / d) * (h / d)))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1e-24)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(Float64(d / M) * Float64(Float64(d / M) / h)) * Float64(l / D)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D / Float64(Float64(l / Float64(M * M)) / Float64(Float64(D / d) * Float64(h / d))))))));
	end
	return tmp
end

MATLAB

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-24)
		tmp = w0 * sqrt((1.0 - (0.25 * (D / (((d / M) * ((d / M) / h)) * (l / D))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D / ((l / (M * M)) / ((D / d) * (h / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

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-24], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(N[(d / M), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D / N[(N[(l / N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(D / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.99999999999999924e-25

   1. Initial program 79.4%

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

      1. times-frac 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in M around 0 47.6%

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

      1. associate-*r/ 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-*r/ 47.6%

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

      4. associate-/l* 46.5%

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

      5. unpow2 46.5%

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

      6. unpow2 46.5%

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

      7. *-commutative 46.5%

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

      8. unpow2 46.5%

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

   6. Simplified 46.5%

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

      1. expm1-log1p-u 31.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 31.8%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 38.5%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 36.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 36.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 36.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 51.7%

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

      3. associate-/l* 51.6%

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

      4. associate-/l* 58.6%

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

   10. Simplified 58.6%

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

       1. associate-/r/ 62.3%

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

   12. Applied egg-rr 62.3%

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

   13. Taylor expanded in l around 0 55.6%

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

       1. *-commutative 55.6%

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

       2. *-commutative 55.6%

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

       3. unpow2 55.6%

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

       4. *-commutative 55.6%

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

       5. associate-*r* 58.5%

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

       6. times-frac 60.1%

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

       7. unpow2 60.1%

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

       8. *-commutative 60.1%

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

       9. times-frac 78.4%

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

       10. associate-/r* 79.6%

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

   15. Simplified 79.6%

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

   if 9.99999999999999924e-25 < d

   1. Initial program 86.4%

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

      1. times-frac 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in M around 0 56.1%

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

      1. associate-*r/ 56.1%

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

      2. *-commutative 56.1%

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

      3. associate-*r/ 56.1%

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

      4. associate-/l* 57.2%

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

      5. unpow2 57.2%

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

      6. unpow2 57.2%

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

      7. *-commutative 57.2%

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

      8. unpow2 57.2%

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

   6. Simplified 57.2%

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

      1. expm1-log1p-u 51.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 51.2%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 59.6%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 63.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 63.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)}} \]
   9. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)}} \]

      2. expm1-log1p 69.3%

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

      3. associate-/l* 70.3%

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

      4. associate-/l* 71.6%

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

   10. Simplified 71.6%

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

       1. associate-/r/ 73.9%

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

   12. Applied egg-rr 73.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\left(\frac{d}{M} \cdot \frac{\frac{d}{M}}{h}\right) \cdot \frac{\ell}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D}{\frac{\frac{\ell}{M \cdot M}}{\frac{D}{d} \cdot \frac{h}{d}}}}\\ \end{array} \]

Alternative 6: 71.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.6 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{M}{\frac{d}{\frac{h \cdot M}{d}}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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 7.6e-39)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (/ M (/ d (/ (* h M) d))))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.6e-39) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M / (d / ((h * M) / d)))))));
	}
	return tmp;
}

Fortran

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 <= 7.6d-39) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * (m / (d_1 / ((h * m) / d_1)))))))
    end if
    code = tmp
end function

Java

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 <= 7.6e-39) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M / (d / ((h * M) / d)))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7.6e-39:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M / (d / ((h * M) / d)))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7.6e-39)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M / Float64(d / Float64(Float64(h * M) / d))))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7.6e-39)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M / (d / ((h * M) / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

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, 7.6e-39], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M / N[(d / N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.6000000000000004e-39

   1. Initial program 85.2%

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

      1. times-frac 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in M around 0 73.8%

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

   if 7.6000000000000004e-39 < M

   1. Initial program 73.9%

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

      1. times-frac 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in M around 0 41.8%

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

      1. *-commutative 41.8%

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

      2. *-commutative 41.8%

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

      3. associate-/l* 43.0%

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

      4. unpow2 43.0%

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

      5. unpow2 43.0%

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

      6. *-commutative 43.0%

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

      7. unpow2 43.0%

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

   6. Simplified 43.0%

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

      1. expm1-log1p-u 26.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 26.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 29.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 34.9%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 34.9%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. expm1-def 34.9%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)} \cdot -0.125\right) \]

      2. expm1-log1p 51.3%

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

      3. associate-/r/ 51.3%

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

      4. unpow2 51.3%

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

      5. unpow2 51.3%

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

      6. times-frac 46.3%

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

      7. *-commutative 46.3%

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

      8. times-frac 49.9%

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

      9. unpow2 49.9%

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

      10. unpow2 49.9%

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

   10. Simplified 49.9%

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

   11. Taylor expanded in D around 0 44.9%

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

       1. unpow2 44.9%

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

       2. times-frac 46.1%

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

       3. unpow2 46.1%

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

       4. associate-*r* 50.1%

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

       5. associate-/l* 62.9%

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

       6. associate-/l* 64.2%

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

       7. *-commutative 64.2%

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

   13. Simplified 64.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.6 \cdot 10^{-39}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{M}{\frac{d}{\frac{h \cdot M}{d}}}\right)\right)\right)\\ \end{array} \]

Alternative 7: 72.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{\frac{D}{d}}{d} \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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-117)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ (/ D d) d) (/ M (/ (/ l h) M)))))))))

C

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-117) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (((D / d) / d) * (M / ((l / h) / M))))));
	}
	return tmp;
}

Fortran

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-117) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * (((d / d_1) / d_1) * (m / ((l / h) / m))))))
    end if
    code = tmp
end function

Java

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-117) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (((D / d) / d) * (M / ((l / h) / M))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.7e-117:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * (((D / d) / d) * (M / ((l / h) / M))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.7e-117)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(Float64(D / d) / d) * Float64(M / Float64(Float64(l / h) / M)))))));
	end
	return tmp
end

MATLAB

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-117)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * (((D / d) / d) * (M / ((l / h) / M))))));
	end
	tmp_2 = tmp;
end

Wolfram

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-117], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(N[(D / d), $MachinePrecision] / d), $MachinePrecision] * N[(M / N[(N[(l / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.70000000000000017e-117

   1. Initial program 85.2%

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

      1. times-frac 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in M around 0 73.4%

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

   if 1.70000000000000017e-117 < M

   1. Initial program 75.4%

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

      1. times-frac 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in M around 0 43.6%

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

      1. *-commutative 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-/l* 44.6%

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

      4. unpow2 44.6%

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

      5. unpow2 44.6%

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

      6. *-commutative 44.6%

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

      7. unpow2 44.6%

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

   6. Simplified 44.6%

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

      1. expm1-log1p-u 29.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

      2. expm1-udef 29.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

      3. associate-/l* 34.3%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

      4. times-frac 38.7%

         \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

   8. Applied egg-rr 38.7%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)} \cdot -0.125\right) \]
   9. Step-by-step derivation

      1. expm1-def 38.7%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)} \cdot -0.125\right) \]

      2. expm1-log1p 54.0%

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

      3. associate-/r/ 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. times-frac 49.7%

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

      7. *-commutative 49.7%

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

      8. times-frac 52.9%

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

      9. unpow2 52.9%

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

      10. unpow2 52.9%

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

   10. Simplified 52.9%

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

   11. Taylor expanded in D around 0 48.5%

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

       1. unpow2 48.5%

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

       2. *-commutative 48.5%

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

       3. times-frac 49.7%

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

       4. associate-/r* 54.0%

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

       5. unpow2 54.0%

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

       6. associate-*r* 59.5%

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

       7. associate-/l* 62.8%

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

       8. associate-/l/ 64.9%

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

   13. Simplified 64.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{\frac{D}{d}}{d} \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 8: 70.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(h \cdot M\right) \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{D}{\ell}}{\frac{d}{w0}}\right)\\ \end{array} \end{array} \]

FPCore

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 2.8e+41)
   w0
   (* -0.125 (* (* (* h M) (/ M d)) (/ (* D (/ D l)) (/ d w0))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.8e+41) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((h * M) * (M / d)) * ((D * (D / l)) / (d / w0)));
	}
	return tmp;
}

Fortran

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 <= 2.8d+41) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((h * m) * (m / d_1)) * ((d * (d / l)) / (d_1 / w0)))
    end if
    code = tmp
end function

Java

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 <= 2.8e+41) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((h * M) * (M / d)) * ((D * (D / l)) / (d / w0)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.8e+41:
		tmp = w0
	else:
		tmp = -0.125 * (((h * M) * (M / d)) * ((D * (D / l)) / (d / w0)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.8e+41)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(h * M) * Float64(M / d)) * Float64(Float64(D * Float64(D / l)) / Float64(d / w0))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.8e+41)
		tmp = w0;
	else
		tmp = -0.125 * (((h * M) * (M / d)) * ((D * (D / l)) / (d / w0)));
	end
	tmp_2 = tmp;
end

Wolfram

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, 2.8e+41], w0, N[(-0.125 * N[(N[(N[(h * M), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(d / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999999e41

   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. times-frac 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in M around 0 73.8%

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

   if 2.7999999999999999e41 < M

   1. Initial program 74.0%

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

      1. times-frac 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in M around 0 31.1%

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

      1. fma-def 31.1%

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

      2. *-commutative 31.1%

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

      3. times-frac 35.2%

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

      4. associate-*r/ 35.2%

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

      5. unpow2 35.2%

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

      6. unpow2 35.2%

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

      7. times-frac 35.8%

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

      8. unpow2 35.8%

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

      9. *-commutative 35.8%

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

      10. *-commutative 35.8%

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

      11. associate-*l* 42.7%

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

      12. unpow2 42.7%

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

   6. Simplified 42.7%

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

      1. expm1-log1p-u 29.9%

         \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{D}{d}\right)}^{2} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right)}{\ell}\right)\right)}, w0\right) \]

      2. expm1-udef 29.9%

         \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\frac{D}{d}\right)}^{2} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right)}{\ell}\right)} - 1}, w0\right) \]

      3. associate-/l* 28.1%

         \[\leadsto \mathsf{fma}\left(-0.125, e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}}}\right)} - 1, w0\right) \]

   8. Applied egg-rr 28.1%

      \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}}\right)} - 1}, w0\right) \]
   9. Step-by-step derivation

      1. expm1-def 28.1%

         \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}}\right)\right)}, w0\right) \]

      2. expm1-log1p 40.9%

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

      3. unpow2 40.9%

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

      4. associate-/r* 42.2%

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

      5. unpow2 42.2%

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

      6. associate-/r* 51.3%

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

   10. Simplified 51.3%

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

   11. Taylor expanded in D around inf 19.1%

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

       1. *-commutative 19.1%

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

       2. unpow2 19.1%

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

       3. unpow2 19.1%

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

       4. *-commutative 19.1%

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

       5. times-frac 19.6%

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

       6. *-commutative 19.6%

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

       7. unpow2 19.6%

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

       8. associate-*r* 19.9%

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

       9. associate-/l* 19.6%

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

   13. Simplified 19.6%

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

   14. Taylor expanded in D around 0 19.1%

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

       1. *-commutative 19.1%

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

       2. times-frac 19.6%

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

       3. *-commutative 19.6%

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

       4. unpow2 19.6%

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

       5. associate-*r* 20.0%

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

       6. unpow2 20.0%

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

       7. times-frac 19.7%

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

       8. associate-*r* 19.9%

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

       9. unpow2 19.9%

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

       10. associate-/l* 20.4%

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

       11. associate-/l* 21.2%

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

       12. associate-*r* 21.0%

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

       13. associate-*r/ 21.4%

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

       14. associate-/l* 20.9%

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

       15. associate-*r/ 21.1%

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

   16. Simplified 21.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(h \cdot M\right) \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{D}{\ell}}{\frac{d}{w0}}\right)\\ \end{array} \]

Alternative 9: 78.3% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. times-frac 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in M around 0 48.7%

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

   1. *-commutative 48.7%

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

   2. *-commutative 48.7%

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

   3. associate-/l* 48.3%

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

   4. unpow2 48.3%

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

   5. unpow2 48.3%

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

   6. *-commutative 48.3%

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

   7. unpow2 48.3%

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

6. Simplified 48.3%

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

   1. expm1-log1p-u 38.3%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)\right)}} \]

   2. expm1-udef 38.3%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D \cdot D}{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}\right)} - 1\right)}} \]

   3. associate-/l* 45.6%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{D}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{\left(M \cdot M\right) \cdot h}}{D}}}\right)} - 1\right)} \]

   4. times-frac 45.2%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\color{blue}{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}}{D}}\right)} - 1\right)} \]

8. Applied egg-rr 45.2%

   \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)} - 1\right)} \cdot -0.125\right) \]
9. Step-by-step derivation

   1. expm1-def 45.2%

      \[\leadsto w0 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\right)} \cdot -0.125\right) \]

   2. expm1-log1p 55.8%

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

   3. associate-/r/ 55.8%

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

   4. unpow2 55.8%

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

   5. unpow2 55.8%

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

   6. times-frac 55.9%

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

   7. *-commutative 55.9%

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

   8. times-frac 56.9%

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

   9. unpow2 56.9%

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

   10. unpow2 56.9%

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

10. Simplified 56.9%

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

11. Taylor expanded in l around 0 55.9%

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

    1. unpow2 55.9%

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

    2. unpow2 55.9%

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

    3. *-rgt-identity 55.9%

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

    4. associate-*r* 59.0%

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

    5. associate-*r/ 58.6%

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

    6. associate-*l* 59.5%

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

    7. associate-*r/ 59.9%

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

    8. *-rgt-identity 59.9%

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

    9. times-frac 74.1%

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

    10. *-commutative 74.1%

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

13. Simplified 74.1%

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

14. Final simplification 74.1%

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

Alternative 10: 67.2% accurate, 216.0× speedup

Math

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

FPCore

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

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

Fortran

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

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

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. times-frac 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in M around 0 69.9%

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

5. Final simplification 69.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023214 
(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))))))