Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 89.1%

Time: 13.3s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% 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: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e+17)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* h (pow (* M (* D (/ 0.5 d))) 2.0)))))))
   (*
    w0
    (sqrt (+ 1.0 (* (* (/ D l) (* (/ h (/ d M)) (/ D (/ d M)))) -0.25))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+17) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * pow((M * (D * (0.5 / d))), 2.0))))));
	} else {
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 5d+17) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0))))))
    else
        tmp = w0 * sqrt((1.0d0 + (((d / l) * ((h / (d_1 / m)) * (d / (d_1 / m)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+17) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (h * Math.pow((M * (D * (0.5 / d))), 2.0))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 5e+17:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (h * math.pow((M * (D * (0.5 / d))), 2.0))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+17)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D / l) * Float64(Float64(h / Float64(d / M)) * Float64(D / Float64(d / M)))) * -0.25))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 5e+17)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (h * ((M * (D * (0.5 / d))) ^ 2.0))))));
	else
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+17], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D / l), $MachinePrecision] * N[(N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5e17

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 83.4%

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

   3. Simplified 83.4%

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

      1. associate-*r/ 93.1%

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

      2. frac-times 93.3%

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

      3. *-commutative 93.3%

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

      4. clear-num 93.3%

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

      5. *-commutative 93.3%

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

      6. div-inv 93.3%

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

      7. associate-*l* 91.7%

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

      8. associate-/r* 91.7%

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

      9. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

   if 5e17 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. times-frac 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in w0 around 0 50.9%

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

      1. *-commutative 50.9%

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

      2. cancel-sign-sub-inv 50.9%

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

      3. *-commutative 50.9%

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

      4. cancel-sign-sub-inv 50.9%

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

      5. *-commutative 50.9%

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

      6. cancel-sign-sub-inv 50.9%

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

      7. distribute-lft-neg-in 50.9%

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

      8. distribute-rgt-neg-in 50.9%

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

   6. Simplified 50.8%

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

      1. frac-times 55.1%

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

      2. *-commutative 55.1%

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

      3. times-frac 76.1%

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

   8. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. times-frac 67.8%

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

      3. associate-/l* 61.9%

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

      4. associate-*l/ 67.8%

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

      5. associate-*l* 74.3%

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

      6. associate-*r/ 76.0%

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

      7. *-commutative 76.0%

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

      8. times-frac 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -4e-13)
   (* w0 (sqrt (+ 1.0 (* (* (/ D l) (* (/ h (/ d M)) (/ D (/ d M)))) -0.25))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-13) {
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-4d-13)) then
        tmp = w0 * sqrt((1.0d0 + (((d / l) * ((h / (d_1 / m)) * (d / (d_1 / m)))) * (-0.25d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-13) {
		tmp = w0 * Math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-13:
		tmp = w0 * math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e-13)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D / l) * Float64(Float64(h / Float64(d / M)) * Float64(D / Float64(d / M)))) * -0.25))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -4e-13)
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e-13], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D / l), $MachinePrecision] * N[(N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.0000000000000001e-13

   1. Initial program 64.3%

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

      1. *-commutative 64.3%

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

      2. times-frac 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in w0 around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. cancel-sign-sub-inv 44.0%

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

      3. *-commutative 44.0%

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

      4. cancel-sign-sub-inv 44.0%

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

      5. *-commutative 44.0%

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

      6. cancel-sign-sub-inv 44.0%

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

      7. distribute-lft-neg-in 44.0%

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

      8. distribute-rgt-neg-in 44.0%

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

   6. Simplified 44.2%

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

      1. frac-times 45.8%

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

      2. *-commutative 45.8%

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

      3. times-frac 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. times-frac 54.3%

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

      3. associate-/l* 51.6%

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

      4. associate-*l/ 54.3%

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

      5. associate-*l* 59.8%

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

      6. associate-*r/ 61.3%

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

      7. *-commutative 61.3%

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

      8. times-frac 63.9%

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

   10. Simplified 63.9%

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

   if -4.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. times-frac 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in M around 0 98.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e+17)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* (* (/ D l) (* (/ h (/ d M)) (/ D (/ d M)))) -0.25))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+17) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((m * d) / (2.0d0 * d_1)) <= 5d+17) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + (((d / l) * ((h / (d_1 / m)) * (d / (d_1 / m)))) * (-0.25d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+17) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 5e+17:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+17)
		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(Float64(Float64(D / l) * Float64(Float64(h / Float64(d / M)) * Float64(D / Float64(d / M)))) * -0.25))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 5e+17)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + (((D / l) * ((h / (d / M)) * (D / (d / M)))) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+17], 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[(N[(N[(D / l), $MachinePrecision] * N[(N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5e17

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 83.4%

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

   3. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. frac-times 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l/ 93.3%

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

      5. div-inv 93.3%

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

      6. associate-*l* 91.7%

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

      7. associate-/r* 91.7%

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

      8. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

   if 5e17 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. times-frac 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in w0 around 0 50.9%

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

      1. *-commutative 50.9%

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

      2. cancel-sign-sub-inv 50.9%

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

      3. *-commutative 50.9%

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

      4. cancel-sign-sub-inv 50.9%

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

      5. *-commutative 50.9%

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

      6. cancel-sign-sub-inv 50.9%

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

      7. distribute-lft-neg-in 50.9%

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

      8. distribute-rgt-neg-in 50.9%

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

   6. Simplified 50.8%

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

      1. frac-times 55.1%

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

      2. *-commutative 55.1%

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

      3. times-frac 76.1%

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

   8. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. times-frac 67.8%

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

      3. associate-/l* 61.9%

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

      4. associate-*l/ 67.8%

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

      5. associate-*l* 74.3%

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

      6. associate-*r/ 76.0%

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

      7. *-commutative 76.0%

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

      8. times-frac 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) \cdot -0.25}\\ \end{array} \]

Alternative 4: 82.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{-206}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.6e-206)
   w0
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ D l) (* (/ h (/ d M)) (* M (/ D d))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.6e-206) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / l) * ((h / (d / M)) * (M * (D / d)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.6d-206) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / l) * ((h / (d_1 / m)) * (m * (d / d_1)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.6e-206) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / l) * ((h / (d / M)) * (M * (D / d)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.6e-206:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / l) * ((h / (d / M)) * (M * (D / d)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.6e-206)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / l) * Float64(Float64(h / Float64(d / M)) * Float64(M * Float64(D / d))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.6e-206)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / l) * ((h / (d / M)) * (M * (D / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.6e-206], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / l), $MachinePrecision] * N[(N[(h / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.59999999999999988e-206

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. times-frac 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in M around 0 79.1%

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

   if 1.59999999999999988e-206 < M

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in w0 around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. cancel-sign-sub-inv 55.9%

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

      3. *-commutative 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. *-commutative 55.9%

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

      6. cancel-sign-sub-inv 55.9%

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

      7. distribute-lft-neg-in 55.9%

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

      8. distribute-rgt-neg-in 55.9%

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

   6. Simplified 61.6%

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

      1. frac-times 64.5%

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

      2. *-commutative 64.5%

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

      3. times-frac 83.4%

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

   8. Applied egg-rr 83.4%

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

      1. *-commutative 83.4%

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

      2. times-frac 77.7%

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

      3. associate-/l* 75.9%

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

      4. associate-*l/ 77.7%

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

      5. associate-*l* 79.8%

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

      6. associate-*r/ 83.4%

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

      7. *-commutative 83.4%

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

      8. times-frac 89.0%

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

   10. Simplified 89.0%

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

       1. associate-/r/ 89.0%

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

   12. Applied egg-rr 89.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{-206}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\ell} \cdot \left(\frac{h}{\frac{d}{M}} \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{h}{\frac{\ell}{D}} \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{M}}, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 3.8e-8)
   (* w0 (fma (* (/ h (/ l D)) (/ D (* (/ d M) (/ d M)))) -0.125 1.0))
   (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3.8e-8) {
		tmp = w0 * fma(((h / (l / D)) * (D / ((d / M) * (d / M)))), -0.125, 1.0);
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 3.8e-8)
		tmp = Float64(w0 * fma(Float64(Float64(h / Float64(l / D)) * Float64(D / Float64(Float64(d / M) * Float64(d / M)))), -0.125, 1.0));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M)))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 3.8e-8], N[(w0 * N[(N[(N[(h / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(D / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3.80000000000000028e-8

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. times-frac 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 52.3%

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

      1. +-commutative 52.3%

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

      2. associate-*r/ 52.3%

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

      3. *-commutative 52.3%

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

      4. associate-*r/ 52.3%

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

      5. *-commutative 52.3%

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

      6. fma-def 52.3%

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

      7. times-frac 49.6%

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

      8. unpow2 49.6%

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

      9. associate-/l* 50.7%

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

      10. associate-/l* 52.9%

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

      11. unpow2 52.9%

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

      12. unpow2 52.9%

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

   6. Simplified 52.9%

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

      1. frac-times 57.8%

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

      2. *-commutative 57.8%

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

      3. times-frac 76.8%

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

   8. Applied egg-rr 72.7%

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

      1. times-frac 72.8%

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

   10. Simplified 72.8%

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

   if 3.80000000000000028e-8 < d

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in M around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. times-frac 66.5%

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

      4. *-commutative 66.5%

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

      5. unpow2 66.5%

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

      6. unpow2 66.5%

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

      7. *-commutative 66.5%

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

      8. unpow2 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in D around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. times-frac 66.5%

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

      3. unpow2 66.5%

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

      4. associate-/l* 63.5%

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

      5. unpow2 63.5%

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

      6. unpow2 63.5%

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

      7. times-frac 69.5%

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

      8. associate-*l* 69.7%

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

      9. associate-/r/ 76.9%

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

      10. *-commutative 76.9%

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

      11. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

       1. associate-*r/ 76.9%

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

   11. Applied egg-rr 76.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{h}{\frac{\ell}{D}} \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{M}}, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 6: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-18}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}}{\frac{\ell}{D}}, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4e-18)
   (* w0 (fma (/ (* D (/ h (* (/ d M) (/ d M)))) (/ l D)) -0.125 1.0))
   (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4e-18) {
		tmp = w0 * fma(((D * (h / ((d / M) * (d / M)))) / (l / D)), -0.125, 1.0);
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4e-18)
		tmp = Float64(w0 * fma(Float64(Float64(D * Float64(h / Float64(Float64(d / M) * Float64(d / M)))) / Float64(l / D)), -0.125, 1.0));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M)))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4e-18], N[(w0 * N[(N[(N[(D * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.0000000000000003e-18

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. times-frac 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 52.3%

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

      1. +-commutative 52.3%

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

      2. associate-*r/ 52.3%

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

      3. *-commutative 52.3%

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

      4. associate-*r/ 52.3%

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

      5. *-commutative 52.3%

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

      6. fma-def 52.3%

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

      7. times-frac 49.6%

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

      8. unpow2 49.6%

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

      9. associate-/l* 50.7%

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

      10. associate-/l* 52.9%

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

      11. unpow2 52.9%

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

      12. unpow2 52.9%

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

   6. Simplified 52.9%

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

      1. associate-*l/ 58.0%

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

      2. times-frac 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 4.0000000000000003e-18 < d

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in M around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. times-frac 66.5%

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

      4. *-commutative 66.5%

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

      5. unpow2 66.5%

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

      6. unpow2 66.5%

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

      7. *-commutative 66.5%

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

      8. unpow2 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in D around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. times-frac 66.5%

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

      3. unpow2 66.5%

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

      4. associate-/l* 63.5%

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

      5. unpow2 63.5%

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

      6. unpow2 63.5%

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

      7. times-frac 69.5%

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

      8. associate-*l* 69.7%

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

      9. associate-/r/ 76.9%

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

      10. *-commutative 76.9%

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

      11. associate-/l* 80.9%

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

   9. Simplified 80.9%

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

       1. associate-*r/ 76.9%

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

   11. Applied egg-rr 76.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-18}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{D \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}}{\frac{\ell}{D}}, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 7: 79.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.35 \cdot 10^{-156}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 0.0001:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{\frac{d}{h}}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.35e-156)
   w0
   (if (<= M 0.0001)
     (* w0 (+ 1.0 (* -0.125 (* (/ D d) (/ (/ (* M (* M D)) (/ d h)) l)))))
     (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.35e-156) {
		tmp = w0;
	} else if (M <= 0.0001) {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * (((M * (M * D)) / (d / h)) / l))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4.35d-156) then
        tmp = w0
    else if (m <= 0.0001d0) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / d_1) * (((m * (m * d)) / (d_1 / h)) / l))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((d / d_1) * ((m * h) / (l / m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.35e-156) {
		tmp = w0;
	} else if (M <= 0.0001) {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * (((M * (M * D)) / (d / h)) / l))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.35e-156:
		tmp = w0
	elif M <= 0.0001:
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * (((M * (M * D)) / (d / h)) / l))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.35e-156)
		tmp = w0;
	elseif (M <= 0.0001)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(Float64(M * Float64(M * D)) / Float64(d / h)) / l)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.35e-156)
		tmp = w0;
	elseif (M <= 0.0001)
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * (((M * (M * D)) / (d / h)) / l))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.35e-156], w0, If[LessEqual[M, 0.0001], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 4.35000000000000005e-156

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 78.6%

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

   if 4.35000000000000005e-156 < M < 1.00000000000000005e-4

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in M around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. times-frac 68.7%

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

      4. *-commutative 68.7%

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

      5. unpow2 68.7%

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

      6. unpow2 68.7%

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

      7. *-commutative 68.7%

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

      8. unpow2 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in D around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. times-frac 68.7%

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

      3. unpow2 68.7%

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

      4. associate-/l* 68.7%

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

      5. unpow2 68.7%

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

      6. unpow2 68.7%

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

      7. times-frac 80.2%

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

      8. associate-*l* 89.0%

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

      9. associate-/r/ 91.8%

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

      10. *-commutative 91.8%

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

      11. associate-/l* 91.8%

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

   9. Simplified 91.8%

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

   10. Taylor expanded in D around 0 86.1%

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

       1. times-frac 89.0%

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

       2. *-commutative 89.0%

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

       3. times-frac 86.1%

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

       4. associate-/l/ 92.0%

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

       5. *-commutative 92.0%

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

       6. associate-*r* 92.0%

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

       7. associate-/l* 94.7%

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

       8. unpow2 94.7%

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

       9. associate-*r* 94.7%

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

   12. Simplified 94.7%

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

   if 1.00000000000000005e-4 < M

   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. *-commutative 83.7%

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

      2. times-frac 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in M around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. *-commutative 52.1%

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

      3. times-frac 47.1%

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

      4. *-commutative 47.1%

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

      5. unpow2 47.1%

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

      6. unpow2 47.1%

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

      7. *-commutative 47.1%

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

      8. unpow2 47.1%

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

   6. Simplified 47.1%

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

   7. Taylor expanded in D around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. times-frac 47.1%

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

      3. unpow2 47.1%

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

      4. associate-/l* 52.1%

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

      5. unpow2 52.1%

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

      6. unpow2 52.1%

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

      7. times-frac 59.2%

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

      8. associate-*l* 61.2%

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

      9. associate-/r/ 59.8%

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

      10. *-commutative 59.8%

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

      11. associate-/l* 67.8%

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

   9. Simplified 67.8%

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

       1. associate-*r/ 66.3%

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

   11. Applied egg-rr 66.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.35 \cdot 10^{-156}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 0.0001:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{\frac{d}{h}}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 8: 78.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.35 \cdot 10^{-156}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.35e-156)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (* h (/ M (/ l M))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.35e-156) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (h * (M / (l / M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4.35d-156) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((d / d_1) * (h * (m / (l / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.35e-156) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (h * (M / (l / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.35e-156:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (h * (M / (l / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.35e-156)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(h * Float64(M / Float64(l / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.35e-156)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * (h * (M / (l / M)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.35e-156], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.35000000000000005e-156

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 78.6%

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

   if 4.35000000000000005e-156 < M

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. times-frac 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in M around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. *-commutative 57.2%

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

      3. times-frac 55.1%

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

      4. *-commutative 55.1%

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

      5. unpow2 55.1%

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

      6. unpow2 55.1%

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

      7. *-commutative 55.1%

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

      8. unpow2 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in D around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. times-frac 55.1%

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

      3. unpow2 55.1%

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

      4. associate-/l* 58.2%

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

      5. unpow2 58.2%

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

      6. unpow2 58.2%

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

      7. times-frac 66.9%

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

      8. associate-*l* 71.4%

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

      9. associate-/r/ 71.6%

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

      10. *-commutative 71.6%

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

      11. associate-/l* 76.6%

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

   9. Simplified 76.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.35 \cdot 10^{-156}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 78.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 5.8e-155)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (/ (* M h) (/ l M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5.8e-155) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 5.8d-155) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((d / d_1) * ((m * h) / (l / m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5.8e-155) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5.8e-155:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5.8e-155)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) / Float64(l / M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5.8e-155)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) / (l / M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5.8e-155], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.80000000000000021e-155

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 78.6%

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

   if 5.80000000000000021e-155 < M

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. times-frac 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in M around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. *-commutative 57.2%

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

      3. times-frac 55.1%

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

      4. *-commutative 55.1%

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

      5. unpow2 55.1%

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

      6. unpow2 55.1%

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

      7. *-commutative 55.1%

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

      8. unpow2 55.1%

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

   6. Simplified 55.1%

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

   7. Taylor expanded in D around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. times-frac 55.1%

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

      3. unpow2 55.1%

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

      4. associate-/l* 58.2%

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

      5. unpow2 58.2%

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

      6. unpow2 58.2%

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

      7. times-frac 66.9%

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

      8. associate-*l* 71.4%

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

      9. associate-/r/ 71.6%

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

      10. *-commutative 71.6%

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

      11. associate-/l* 76.6%

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

   9. Simplified 76.6%

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

       1. associate-*r/ 75.7%

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

   11. Applied egg-rr 75.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 10: 72.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 2.3e+56)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 (* h (* M M))) l)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.3e+56) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.3d+56) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * (h * (m * m))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.3e+56) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.3e+56:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.3e+56)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.3e+56)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (h * (M * M))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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, 2.3e+56], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.30000000000000015e56

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in M around 0 79.7%

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

   if 2.30000000000000015e56 < M

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. times-frac 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in M around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. times-frac 45.2%

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

      4. *-commutative 45.2%

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

      5. unpow2 45.2%

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

      6. unpow2 45.2%

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

      7. *-commutative 45.2%

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

      8. unpow2 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in D around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. times-frac 45.2%

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

      3. unpow2 45.2%

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

      4. associate-/l* 49.3%

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

      5. unpow2 49.3%

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

      6. unpow2 49.3%

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

      7. times-frac 56.0%

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

      8. associate-*l* 58.3%

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

      9. associate-/r/ 58.6%

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

      10. *-commutative 58.6%

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

      11. associate-/l* 68.4%

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

   9. Simplified 68.4%

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

   10. Taylor expanded in D around inf 43.4%

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

       1. associate-*r/ 43.4%

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

       2. *-commutative 43.4%

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

       3. *-commutative 43.4%

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

       4. associate-*r/ 43.4%

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

       5. times-frac 41.3%

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

       6. unpow2 41.3%

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

       7. unpow2 41.3%

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

       8. *-commutative 41.3%

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

       9. unpow2 41.3%

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

   12. Simplified 41.3%

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

   13. Taylor expanded in D around 0 41.3%

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

       1. unpow2 41.3%

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

       2. unpow2 41.3%

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

       3. times-frac 45.9%

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

   15. Simplified 45.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 11: 67.6% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. *-commutative 81.4%

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

   2. times-frac 80.9%

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

3. Simplified 80.9%

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

4. Taylor expanded in M around 0 72.5%

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

5. Final simplification 72.5%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023227 
(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))))))