Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.5%

Time: 15.8s

Alternatives: 7

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.9× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
   (* (/ (sqrt (* -0.25 (/ (* D_m (* (* M_m h) (* M_m D_m))) l))) d_m) w0)
   (*
    w0
    (sqrt
     (+
      1.0
      (* h (/ (/ (* D_m (/ M_m (/ d_m (* M_m D_m)))) (* d_m -4.0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = (sqrt((-0.25 * ((D_m * ((M_m * h) * (M_m * D_m))) / l))) / d_m) * w0;
	} else {
		tmp = w0 * sqrt((1.0 + (h * (((D_m * (M_m / (d_m / (M_m * D_m)))) / (d_m * -4.0)) / l))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-0.25 * ((D_m * ((M_m * h) * (M_m * D_m))) / l))) / d_m) * w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (h * (((D_m * (M_m / (d_m / (M_m * D_m)))) / (d_m * -4.0)) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -math.inf:
		tmp = (math.sqrt((-0.25 * ((D_m * ((M_m * h) * (M_m * D_m))) / l))) / d_m) * w0
	else:
		tmp = w0 * math.sqrt((1.0 + (h * (((D_m * (M_m / (d_m / (M_m * D_m)))) / (d_m * -4.0)) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(-0.25 * Float64(Float64(D_m * Float64(Float64(M_m * h) * Float64(M_m * D_m))) / l))) / d_m) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(h * Float64(Float64(Float64(D_m * Float64(M_m / Float64(d_m / Float64(M_m * D_m)))) / Float64(d_m * -4.0)) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = (sqrt((-0.25 * ((D_m * ((M_m * h) * (M_m * D_m))) / l))) / d_m) * w0;
	else
		tmp = w0 * sqrt((1.0 + (h * (((D_m * (M_m / (d_m / (M_m * D_m)))) / (d_m * -4.0)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(-0.25 * N[(N[(D$95$m * N[(N[(M$95$m * h), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d$95$m), $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(h * N[(N[(N[(D$95$m * N[(M$95$m / N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * -4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 44.2%

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

   3. Simplified 45.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 39.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   7. Simplified 39.8%

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \cdot \color{blue}{w0} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{\frac{-1}{4} \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right), \color{blue}{w0}\right) \]

   9. Applied egg-rr 19.3%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot D\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\left(h \cdot M\right) \cdot M\right) \cdot D\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(h \cdot M\right) \cdot \left(M \cdot D\right)\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(h \cdot M\right) \cdot \left(D \cdot M\right)\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(h \cdot M\right), \left(D \cdot M\right)\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(D \cdot M\right)\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

       7. *-lowering-*.f64 27.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{*.f64}\left(D, M\right)\right)\right), \ell\right)\right)\right), d\right), w0\right) \]

   11. Applied egg-rr 27.8%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 93.3%

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

   3. Simplified 96.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(h \cdot \frac{\frac{D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \left(\frac{\frac{D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{D \cdot \left(M \cdot \frac{M \cdot D}{d}\right)}{-4}}{d}\right), \ell\right)\right)\right)\right)\right) \]

      6. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{D \cdot \left(M \cdot \frac{M \cdot D}{d}\right)}{d \cdot -4}\right), \ell\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(M \cdot \frac{M \cdot D}{d}\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(M \cdot \frac{M \cdot D}{d}\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(M \cdot \frac{1}{\frac{d}{M \cdot D}}\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{M}{\frac{d}{M \cdot D}}\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(M, \left(\frac{d}{M \cdot D}\right)\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \left(D \cdot M\right)\right)\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), \left(d \cdot -4\right)\right), \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 97.5%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \ell\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.5%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{h \cdot \frac{\frac{D \cdot \frac{M}{\frac{d}{D \cdot M}}}{d \cdot -4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

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

Alternative 2: 79.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left(M\_m \cdot D\_m\right)\\ \mathbf{if}\;D\_m \leq 5 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\frac{\frac{-0.125}{\frac{\frac{\ell}{D\_m}}{M\_m \cdot t\_0}}}{d\_m}}{d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{t\_0}{\frac{d\_m}{M\_m \cdot D\_m}}}{d\_m \cdot \ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* h (* M_m D_m))))
   (if (<= D_m 5e-13)
     (* w0 (+ 1.0 (/ (/ (/ -0.125 (/ (/ l D_m) (* M_m t_0))) d_m) d_m)))
     (* w0 (+ 1.0 (* -0.125 (/ (/ t_0 (/ d_m (* M_m D_m))) (* d_m l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = h * (M_m * D_m);
	double tmp;
	if (D_m <= 5e-13) {
		tmp = w0 * (1.0 + (((-0.125 / ((l / D_m) / (M_m * t_0))) / d_m) / d_m));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((t_0 / (d_m / (M_m * D_m))) / (d_m * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = h * (m_m * d_m)
    if (d_m <= 5d-13) then
        tmp = w0 * (1.0d0 + ((((-0.125d0) / ((l / d_m) / (m_m * t_0))) / d_m_1) / d_m_1))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((t_0 / (d_m_1 / (m_m * d_m))) / (d_m_1 * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = h * (M_m * D_m);
	double tmp;
	if (D_m <= 5e-13) {
		tmp = w0 * (1.0 + (((-0.125 / ((l / D_m) / (M_m * t_0))) / d_m) / d_m));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((t_0 / (d_m / (M_m * D_m))) / (d_m * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = h * (M_m * D_m)
	tmp = 0
	if D_m <= 5e-13:
		tmp = w0 * (1.0 + (((-0.125 / ((l / D_m) / (M_m * t_0))) / d_m) / d_m))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((t_0 / (d_m / (M_m * D_m))) / (d_m * l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(h * Float64(M_m * D_m))
	tmp = 0.0
	if (D_m <= 5e-13)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(-0.125 / Float64(Float64(l / D_m) / Float64(M_m * t_0))) / d_m) / d_m)));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(t_0 / Float64(d_m / Float64(M_m * D_m))) / Float64(d_m * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = h * (M_m * D_m);
	tmp = 0.0;
	if (D_m <= 5e-13)
		tmp = w0 * (1.0 + (((-0.125 / ((l / D_m) / (M_m * t_0))) / d_m) / d_m));
	else
		tmp = w0 * (1.0 + (-0.125 * ((t_0 / (d_m / (M_m * D_m))) / (d_m * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 5e-13], N[(w0 * N[(1.0 + N[(N[(N[(-0.125 / N[(N[(l / D$95$m), $MachinePrecision] / N[(M$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(t$95$0 / N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 4.9999999999999999e-13

   1. Initial program 84.6%

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

   3. Simplified 87.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 61.1%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 61.1%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\ell}}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\ell}}{d}}{\color{blue}{d}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\ell}}{d}\right), \color{blue}{d}\right)\right)\right) \]

   9. Applied egg-rr 79.1%

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

   if 4.9999999999999999e-13 < D

   1. Initial program 72.8%

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

   3. Simplified 74.8%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 58.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 58.4%

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

      1. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \color{blue}{\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot M\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{D \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}}{\color{blue}{d \cdot \ell}}\right)\right)\right)\right) \]

   9. Applied egg-rr 67.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-13}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\frac{\frac{-0.125}{\frac{\frac{\ell}{D}}{M \cdot \left(h \cdot \left(M \cdot D\right)\right)}}}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{h \cdot \left(M \cdot D\right)}{\frac{d}{M \cdot D}}}{d \cdot \ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{h \cdot \left(M\_m \cdot D\_m\right)}{\frac{d\_m}{M\_m \cdot D\_m}}}{d\_m \cdot \ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 1.05e-38)
   w0
   (*
    w0
    (+
     1.0
     (* -0.125 (/ (/ (* h (* M_m D_m)) (/ d_m (* M_m D_m))) (* d_m l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.05e-38) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((h * (M_m * D_m)) / (d_m / (M_m * D_m))) / (d_m * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 1.05d-38) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((h * (m_m * d_m)) / (d_m_1 / (m_m * d_m))) / (d_m_1 * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.05e-38) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((h * (M_m * D_m)) / (d_m / (M_m * D_m))) / (d_m * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 1.05e-38:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((h * (M_m * D_m)) / (d_m / (M_m * D_m))) / (d_m * l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.05e-38)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(h * Float64(M_m * D_m)) / Float64(d_m / Float64(M_m * D_m))) / Float64(d_m * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 1.05e-38)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((h * (M_m * D_m)) / (d_m / (M_m * D_m))) / (d_m * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 1.05e-38], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.05000000000000006e-38

   1. Initial program 84.9%

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

   3. Simplified 87.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 77.4%

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

   if 1.05000000000000006e-38 < D

   1. Initial program 73.5%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 58.2%

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

      1. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \color{blue}{\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot \frac{D \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot M\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{D \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(\left(h \cdot M\right) \cdot M\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right)\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d}}{\color{blue}{d \cdot \ell}}\right)\right)\right)\right) \]

   9. Applied egg-rr 69.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\frac{h \cdot \left(M \cdot D\right)}{\frac{d}{M \cdot D}}}{d \cdot \ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\frac{-0.125}{d\_m} \cdot \left(M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot \left(D\_m \cdot h\right)\right)\right)\right)}{d\_m \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 4e-5)
   w0
   (* w0 (/ (* (/ -0.125 d_m) (* M_m (* D_m (* M_m (* D_m h))))) (* d_m l)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = w0 * (((-0.125 / d_m) * (M_m * (D_m * (M_m * (D_m * h))))) / (d_m * l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 4d-5) then
        tmp = w0
    else
        tmp = w0 * ((((-0.125d0) / d_m_1) * (m_m * (d_m * (m_m * (d_m * h))))) / (d_m_1 * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = w0 * (((-0.125 / d_m) * (M_m * (D_m * (M_m * (D_m * h))))) / (d_m * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 4e-5:
		tmp = w0
	else:
		tmp = w0 * (((-0.125 / d_m) * (M_m * (D_m * (M_m * (D_m * h))))) / (d_m * l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(-0.125 / d_m) * Float64(M_m * Float64(D_m * Float64(M_m * Float64(D_m * h))))) / Float64(d_m * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = w0 * (((-0.125 / d_m) * (M_m * (D_m * (M_m * (D_m * h))))) / (d_m * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 4e-5], w0, N[(w0 * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(M$95$m * N[(D$95$m * N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000033e-5

   1. Initial program 82.8%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 78.1%

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

   if 4.00000000000000033e-5 < M

   1. Initial program 78.6%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 47.8%

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

   8. Taylor expanded in d around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \ell\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left(d \cdot d\right)\right)\right), \ell\right)\right) \]

      15. *-lowering-*.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right)\right) \]

   10. Simplified 23.8%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}}{\ell}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{d \cdot d}}{\ell}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{d \cdot d}}{\ell}\right)\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8}}{d \cdot d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{\ell}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8}}{d \cdot d} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\ell}}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8}}{d}}{d} \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}}{\ell}\right)\right) \]

       7. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8}}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{\color{blue}{d \cdot \ell}}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8}}{d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)\right), \color{blue}{\left(d \cdot \ell\right)}\right)\right) \]

   12. Applied egg-rr 27.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\frac{-0.125}{d} \cdot \left(M \cdot \left(D \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)\right)}{d \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{-0.125}{\frac{\ell}{M\_m \cdot \left(D\_m \cdot \left(M\_m \cdot \left(D\_m \cdot h\right)\right)\right)} \cdot \left(d\_m \cdot d\_m\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3.8e-5)
   w0
   (* w0 (/ -0.125 (* (/ l (* M_m (* D_m (* M_m (* D_m h))))) (* d_m d_m))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.8e-5) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 / ((l / (M_m * (D_m * (M_m * (D_m * h))))) * (d_m * d_m)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 3.8d-5) then
        tmp = w0
    else
        tmp = w0 * ((-0.125d0) / ((l / (m_m * (d_m * (m_m * (d_m * h))))) * (d_m_1 * d_m_1)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.8e-5) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 / ((l / (M_m * (D_m * (M_m * (D_m * h))))) * (d_m * d_m)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3.8e-5:
		tmp = w0
	else:
		tmp = w0 * (-0.125 / ((l / (M_m * (D_m * (M_m * (D_m * h))))) * (d_m * d_m)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3.8e-5)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(-0.125 / Float64(Float64(l / Float64(M_m * Float64(D_m * Float64(M_m * Float64(D_m * h))))) * Float64(d_m * d_m))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3.8e-5)
		tmp = w0;
	else
		tmp = w0 * (-0.125 / ((l / (M_m * (D_m * (M_m * (D_m * h))))) * (d_m * d_m)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3.8e-5], w0, N[(w0 * N[(-0.125 / N[(N[(l / N[(M$95$m * N[(D$95$m * N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.8000000000000002e-5

   1. Initial program 82.8%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 78.1%

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

   if 3.8000000000000002e-5 < M

   1. Initial program 78.6%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 47.8%

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

   8. Taylor expanded in d around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \ell\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left(d \cdot d\right)\right)\right), \ell\right)\right) \]

      15. *-lowering-*.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right)\right) \]

   10. Simplified 23.8%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}}{\ell}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{d \cdot d}}{\ell}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{d \cdot d}}{\ell}\right)\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{\frac{-1}{8}}{d \cdot d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{\ell}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot \frac{\frac{-1}{8}}{d \cdot d}}{\ell}\right)\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\ell} \cdot \color{blue}{\frac{\frac{-1}{8}}{d \cdot d}}\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{1}{\frac{\ell}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}} \cdot \frac{\color{blue}{\frac{-1}{8}}}{d \cdot d}\right)\right) \]

       8. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{1 \cdot \frac{-1}{8}}{\color{blue}{\frac{\ell}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h} \cdot \left(d \cdot d\right)}}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8}}{\color{blue}{\frac{\ell}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}} \cdot \left(d \cdot d\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\frac{-1}{8}, \color{blue}{\left(\frac{\ell}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h} \cdot \left(d \cdot d\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(\frac{\ell}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}\right), \color{blue}{\left(d \cdot d\right)}\right)\right)\right) \]

   12. Applied egg-rr 24.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{-0.125}{\frac{\ell}{M \cdot \left(D \cdot \left(M \cdot \left(D \cdot h\right)\right)\right)} \cdot \left(d \cdot d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-5}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;M\_m \cdot \left(\frac{M\_m \cdot \left(D\_m \cdot h\right)}{\frac{\ell}{D\_m}} \cdot \left(w0 \cdot \frac{-0.125}{d\_m \cdot d\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 4e-5)
   w0
   (* M_m (* (/ (* M_m (* D_m h)) (/ l D_m)) (* w0 (/ -0.125 (* d_m d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = M_m * (((M_m * (D_m * h)) / (l / D_m)) * (w0 * (-0.125 / (d_m * d_m))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 4d-5) then
        tmp = w0
    else
        tmp = m_m * (((m_m * (d_m * h)) / (l / d_m)) * (w0 * ((-0.125d0) / (d_m_1 * d_m_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4e-5) {
		tmp = w0;
	} else {
		tmp = M_m * (((M_m * (D_m * h)) / (l / D_m)) * (w0 * (-0.125 / (d_m * d_m))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 4e-5:
		tmp = w0
	else:
		tmp = M_m * (((M_m * (D_m * h)) / (l / D_m)) * (w0 * (-0.125 / (d_m * d_m))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = Float64(M_m * Float64(Float64(Float64(M_m * Float64(D_m * h)) / Float64(l / D_m)) * Float64(w0 * Float64(-0.125 / Float64(d_m * d_m)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 4e-5)
		tmp = w0;
	else
		tmp = M_m * (((M_m * (D_m * h)) / (l / D_m)) * (w0 * (-0.125 / (d_m * d_m))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 4e-5], w0, N[(M$95$m * N[(N[(N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(l / D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(-0.125 / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000033e-5

   1. Initial program 82.8%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 78.1%

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

   if 4.00000000000000033e-5 < M

   1. Initial program 78.6%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8}}{{d}^{2}} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{{d}^{2}}\right), \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({d}^{2}\right)\right), \left(\frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left(d \cdot d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \ell\right)\right)\right)\right) \]

   7. Simplified 47.8%

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

   8. Taylor expanded in d around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right), \color{blue}{\ell}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right), \ell\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right), \ell\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left({d}^{2}\right)\right)\right), \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \left(d \cdot d\right)\right)\right), \ell\right)\right) \]

      15. *-lowering-*.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right)\right) \]

   10. Simplified 23.8%

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

       1. associate-*r/ N/A

          \[\leadsto \frac{w0 \cdot \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}{\color{blue}{\ell}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{w0 \cdot \frac{\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}}{\ell} \]

       3. associate-*r* N/A

          \[\leadsto \frac{w0 \cdot \frac{\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{d \cdot d}}{\ell} \]

       4. associate-*l* N/A

          \[\leadsto \frac{w0 \cdot \frac{\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{d \cdot d}}{\ell} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{w0 \cdot \left(\frac{\frac{-1}{8}}{d \cdot d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)\right)}{\ell} \]

       6. associate-*r/ N/A

          \[\leadsto w0 \cdot \color{blue}{\frac{\frac{\frac{-1}{8}}{d \cdot d} \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h\right)}{\ell}} \]

       7. associate-*r/ N/A

          \[\leadsto w0 \cdot \left(\frac{\frac{-1}{8}}{d \cdot d} \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\ell}}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(w0 \cdot \frac{\frac{-1}{8}}{d \cdot d}\right) \cdot \color{blue}{\frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{\frac{-1}{8}}{d \cdot d} \cdot w0\right) \cdot \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}}{\ell} \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot h}{\ell} \cdot \color{blue}{\left(\frac{\frac{-1}{8}}{d \cdot d} \cdot w0\right)} \]

   12. Applied egg-rr 24.7%

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

Alternative 7: 67.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 81.8%

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

3. Simplified 84.3%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 71.0%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))))