Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 87.8%

Time: 17.7s

Alternatives: 18

Speedup: 1.7×

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: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{hypot}\left(1, {d\_m}^{-0.5} \cdot {\left(\frac{\ell}{\frac{M \cdot \frac{M \cdot D\_m}{d\_m}}{\frac{-4}{D\_m \cdot h}}}\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 5e+278)
     (* w0 (sqrt t_0))
     (*
      w0
      (hypot
       1.0
       (*
        (pow d_m -0.5)
        (pow (/ l (/ (* M (/ (* M D_m) d_m)) (/ -4.0 (* D_m h)))) -0.5)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = 1.0 - (pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * hypot(1.0, (pow(d_m, -0.5) * pow((l / ((M * ((M * D_m) / d_m)) / (-4.0 / (D_m * h)))), -0.5)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = 1.0 - (Math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.hypot(1.0, (Math.pow(d_m, -0.5) * Math.pow((l / ((M * ((M * D_m) / d_m)) / (-4.0 / (D_m * h)))), -0.5)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	t_0 = 1.0 - (math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+278:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.hypot(1.0, (math.pow(d_m, -0.5) * math.pow((l / ((M * ((M * D_m) / d_m)) / (-4.0 / (D_m * h)))), -0.5)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+278)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * hypot(1.0, Float64((d_m ^ -0.5) * (Float64(l / Float64(Float64(M * Float64(Float64(M * D_m) / d_m)) / Float64(-4.0 / Float64(D_m * h)))) ^ -0.5))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	t_0 = 1.0 - ((((M * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+278)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * hypot(1.0, ((d_m ^ -0.5) * ((l / ((M * ((M * D_m) / d_m)) / (-4.0 / (D_m * h)))) ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+278], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[1.0 ^ 2 + N[(N[Power[d$95$m, -0.5], $MachinePrecision] * N[Power[N[(l / N[(N[(M * N[(N[(M * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / N[(-4.0 / N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5.00000000000000029e278

   1. Initial program 99.9%

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

   if 5.00000000000000029e278 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 46.4%

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

   3. Simplified 66.0%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. /-lowering-/.f64 64.3%

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

   6. Applied egg-rr 64.3%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{h \cdot \frac{\frac{M}{\frac{d}{D \cdot M}}}{-4}}{d} \cdot \frac{D}{\ell}}} \]
   7. Step-by-step derivation

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

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

      10. hypot-1-def N/A

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

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

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

   8. Applied egg-rr 64.0%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{hypot}\left(1, {\left(\frac{d}{\frac{h \cdot D}{-4}} \cdot \frac{\ell}{\frac{M \cdot \left(M \cdot D\right)}{d}}\right)}^{-0.5}\right)} \]
   9. Step-by-step derivation

      1. frac-times N/A

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

      2. associate-/l* N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

   10. Applied egg-rr 35.4%

       \[\leadsto w0 \cdot \mathsf{hypot}\left(1, \color{blue}{{d}^{-0.5} \cdot {\left(\frac{\ell}{\frac{M \cdot \frac{D \cdot M}{d}}{\frac{-4}{h \cdot D}}}\right)}^{-0.5}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{hypot}\left(1, {d}^{-0.5} \cdot {\left(\frac{\ell}{\frac{M \cdot \frac{M \cdot D}{d}}{\frac{-4}{D \cdot h}}}\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+258}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot \left(M \cdot D\_m\right)\right) \cdot \frac{M \cdot D\_m}{d\_m \cdot -4}}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 2e+258)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt
       (+
        1.0
        (/ (/ (* (* h (* M D_m)) (/ (* M D_m) (* d_m -4.0))) d_m) l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = 1.0 - (pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+258) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 = 1.0d0 - ((((m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 2d+258) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 + ((((h * (m * d_m)) * ((m * d_m) / (d_m_1 * (-4.0d0)))) / d_m_1) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = 1.0 - (Math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+258) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	t_0 = 1.0 - (math.pow(((M * D_m) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 2e+258:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2e+258)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(h * Float64(M * D_m)) * Float64(Float64(M * D_m) / Float64(d_m * -4.0))) / d_m) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	t_0 = 1.0 - ((((M * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 2e+258)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+258], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(h * N[(M * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D$95$m), $MachinePrecision] / N[(d$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 2.00000000000000011e258

   1. Initial program 99.9%

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

   if 2.00000000000000011e258 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.6%

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

   3. Simplified 65.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-/l/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 69.9%

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

   6. Applied egg-rr 69.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 72.2%

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

   8. Applied egg-rr 72.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \frac{M \cdot D}{d \cdot -4}}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-63}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d\_m}{M \cdot D\_m}}}{-4}}{d\_m} \cdot \frac{D\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot \left(M \cdot D\_m\right)\right) \cdot \frac{M \cdot D\_m}{d\_m \cdot -4}}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= l -4e-63)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* h (/ (/ M (/ d_m (* M D_m))) -4.0)) d_m) (/ D_m l)))))
   (*
    w0
    (sqrt
     (+ 1.0 (/ (/ (* (* h (* M D_m)) (/ (* M D_m) (* d_m -4.0))) d_m) l))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -4e-63) {
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 (l <= (-4d-63)) then
        tmp = w0 * sqrt((1.0d0 + (((h * ((m / (d_m_1 / (m * d_m))) / (-4.0d0))) / d_m_1) * (d_m / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((((h * (m * d_m)) * ((m * d_m) / (d_m_1 * (-4.0d0)))) / d_m_1) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -4e-63) {
		tmp = w0 * Math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if l <= -4e-63:
		tmp = w0 * math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (l <= -4e-63)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M / Float64(d_m / Float64(M * D_m))) / -4.0)) / d_m) * Float64(D_m / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(h * Float64(M * D_m)) * Float64(Float64(M * D_m) / Float64(d_m * -4.0))) / d_m) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (l <= -4e-63)
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	else
		tmp = w0 * sqrt((1.0 + ((((h * (M * D_m)) * ((M * D_m) / (d_m * -4.0))) / d_m) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[l, -4e-63], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M / N[(d$95$m / N[(M * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(h * N[(M * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D$95$m), $MachinePrecision] / N[(d$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.00000000000000027e-63

   1. Initial program 89.5%

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

   3. Simplified 87.1%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. /-lowering-/.f64 86.9%

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

   6. Applied egg-rr 86.9%

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

   if -4.00000000000000027e-63 < l

   1. Initial program 79.4%

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

   3. Simplified 83.8%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-/l/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 89.1%

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

   8. Applied egg-rr 89.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-63}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d}{M \cdot D}}}{-4}}{d} \cdot \frac{D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \frac{M \cdot D}{d \cdot -4}}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d\_m}{M \cdot D\_m}}}{-4}}{d\_m} \cdot \frac{D\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D\_m \cdot \frac{M \cdot \frac{M \cdot D\_m}{d\_m}}{-4}\right)}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= l -3.6e-90)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* h (/ (/ M (/ d_m (* M D_m))) -4.0)) d_m) (/ D_m l)))))
   (*
    w0
    (sqrt
     (+ 1.0 (/ (/ (* h (* D_m (/ (* M (/ (* M D_m) d_m)) -4.0))) d_m) l))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -3.6e-90) {
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (((h * (D_m * ((M * ((M * D_m) / d_m)) / -4.0))) / d_m) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 (l <= (-3.6d-90)) then
        tmp = w0 * sqrt((1.0d0 + (((h * ((m / (d_m_1 / (m * d_m))) / (-4.0d0))) / d_m_1) * (d_m / l))))
    else
        tmp = w0 * sqrt((1.0d0 + (((h * (d_m * ((m * ((m * d_m) / d_m_1)) / (-4.0d0)))) / d_m_1) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -3.6e-90) {
		tmp = w0 * Math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h * (D_m * ((M * ((M * D_m) / d_m)) / -4.0))) / d_m) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if l <= -3.6e-90:
		tmp = w0 * math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((h * (D_m * ((M * ((M * D_m) / d_m)) / -4.0))) / d_m) / l)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (l <= -3.6e-90)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M / Float64(d_m / Float64(M * D_m))) / -4.0)) / d_m) * Float64(D_m / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(D_m * Float64(Float64(M * Float64(Float64(M * D_m) / d_m)) / -4.0))) / d_m) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (l <= -3.6e-90)
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	else
		tmp = w0 * sqrt((1.0 + (((h * (D_m * ((M * ((M * D_m) / d_m)) / -4.0))) / d_m) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[l, -3.6e-90], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M / N[(d$95$m / N[(M * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(D$95$m * N[(N[(M * N[(N[(M * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.59999999999999981e-90

   1. Initial program 87.8%

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

   3. Simplified 85.4%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. /-lowering-/.f64 85.3%

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

   6. Applied egg-rr 85.3%

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

   if -3.59999999999999981e-90 < l

   1. Initial program 80.0%

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

   3. Simplified 84.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-90}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d}{M \cdot D}}}{-4}}{d} \cdot \frac{D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 10^{+60}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d\_m}{M \cdot D\_m}}}{-4}}{d\_m} \cdot \frac{D\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D\_m \cdot -0.125\right) \cdot \left(D\_m \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= d_m 1e+60)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* h (/ (/ M (/ d_m (* M D_m))) -4.0)) d_m) (/ D_m l)))))
   (+
    w0
    (* (* D_m -0.125) (* D_m (/ (* w0 (* h (* M M))) (* d_m (* d_m l))))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1e+60) {
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 + ((D_m * -0.125) * (D_m * ((w0 * (h * (M * M))) / (d_m * (d_m * l)))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 1d+60) then
        tmp = w0 * sqrt((1.0d0 + (((h * ((m / (d_m_1 / (m * d_m))) / (-4.0d0))) / d_m_1) * (d_m / l))))
    else
        tmp = w0 + ((d_m * (-0.125d0)) * (d_m * ((w0 * (h * (m * m))) / (d_m_1 * (d_m_1 * l)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1e+60) {
		tmp = w0 * Math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	} else {
		tmp = w0 + ((D_m * -0.125) * (D_m * ((w0 * (h * (M * M))) / (d_m * (d_m * l)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 1e+60:
		tmp = w0 * math.sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))))
	else:
		tmp = w0 + ((D_m * -0.125) * (D_m * ((w0 * (h * (M * M))) / (d_m * (d_m * l)))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 1e+60)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M / Float64(d_m / Float64(M * D_m))) / -4.0)) / d_m) * Float64(D_m / l)))));
	else
		tmp = Float64(w0 + Float64(Float64(D_m * -0.125) * Float64(D_m * Float64(Float64(w0 * Float64(h * Float64(M * M))) / Float64(d_m * Float64(d_m * l))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 1e+60)
		tmp = w0 * sqrt((1.0 + (((h * ((M / (d_m / (M * D_m))) / -4.0)) / d_m) * (D_m / l))));
	else
		tmp = w0 + ((D_m * -0.125) * (D_m * ((w0 * (h * (M * M))) / (d_m * (d_m * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 1e+60], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M / N[(d$95$m / N[(M * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 + N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(D$95$m * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.9999999999999995e59

   1. Initial program 82.6%

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

   3. Simplified 82.9%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. /-lowering-/.f64 81.2%

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

   6. Applied egg-rr 81.2%

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

   if 9.9999999999999995e59 < d

   1. Initial program 82.6%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 56.3%

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

   7. Simplified 56.3%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 72.4%

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

   9. Applied egg-rr 72.4%

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

   10. Taylor expanded in M around 0

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

       1. associate-*r* N/A

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

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

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 63.7%

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

   12. Simplified 63.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{+60}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d}{M \cdot D}}}{-4}}{d} \cdot \frac{D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \left(D \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{-0.125}{d\_m \cdot d\_m}\\ \mathbf{if}\;M \leq 5.9 \cdot 10^{-130}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-21}:\\ \;\;\;\;w0 \cdot \left(1 + t\_0 \cdot \frac{D\_m \cdot \left(h \cdot \left(D\_m \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;M \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + t\_0 \cdot \frac{D\_m \cdot \left(D\_m \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ -0.125 (* d_m d_m))))
   (if (<= M 5.9e-130)
     w0
     (if (<= M 2.2e-21)
       (* w0 (+ 1.0 (* t_0 (/ (* D_m (* h (* D_m (* M M)))) l))))
       (if (<= M 1.6e+36)
         w0
         (* w0 (+ 1.0 (* t_0 (/ (* D_m (* D_m (* h (* M M)))) l)))))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = -0.125 / (d_m * d_m);
	double tmp;
	if (M <= 5.9e-130) {
		tmp = w0;
	} else if (M <= 2.2e-21) {
		tmp = w0 * (1.0 + (t_0 * ((D_m * (h * (D_m * (M * M)))) / l)));
	} else if (M <= 1.6e+36) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (t_0 * ((D_m * (D_m * (h * (M * M)))) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 = (-0.125d0) / (d_m_1 * d_m_1)
    if (m <= 5.9d-130) then
        tmp = w0
    else if (m <= 2.2d-21) then
        tmp = w0 * (1.0d0 + (t_0 * ((d_m * (h * (d_m * (m * m)))) / l)))
    else if (m <= 1.6d+36) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (t_0 * ((d_m * (d_m * (h * (m * m)))) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = -0.125 / (d_m * d_m);
	double tmp;
	if (M <= 5.9e-130) {
		tmp = w0;
	} else if (M <= 2.2e-21) {
		tmp = w0 * (1.0 + (t_0 * ((D_m * (h * (D_m * (M * M)))) / l)));
	} else if (M <= 1.6e+36) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (t_0 * ((D_m * (D_m * (h * (M * M)))) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	t_0 = -0.125 / (d_m * d_m)
	tmp = 0
	if M <= 5.9e-130:
		tmp = w0
	elif M <= 2.2e-21:
		tmp = w0 * (1.0 + (t_0 * ((D_m * (h * (D_m * (M * M)))) / l)))
	elif M <= 1.6e+36:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (t_0 * ((D_m * (D_m * (h * (M * M)))) / l)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	t_0 = Float64(-0.125 / Float64(d_m * d_m))
	tmp = 0.0
	if (M <= 5.9e-130)
		tmp = w0;
	elseif (M <= 2.2e-21)
		tmp = Float64(w0 * Float64(1.0 + Float64(t_0 * Float64(Float64(D_m * Float64(h * Float64(D_m * Float64(M * M)))) / l))));
	elseif (M <= 1.6e+36)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(t_0 * Float64(Float64(D_m * Float64(D_m * Float64(h * Float64(M * M)))) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	t_0 = -0.125 / (d_m * d_m);
	tmp = 0.0;
	if (M <= 5.9e-130)
		tmp = w0;
	elseif (M <= 2.2e-21)
		tmp = w0 * (1.0 + (t_0 * ((D_m * (h * (D_m * (M * M)))) / l)));
	elseif (M <= 1.6e+36)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (t_0 * ((D_m * (D_m * (h * (M * M)))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(-0.125 / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 5.9e-130], w0, If[LessEqual[M, 2.2e-21], N[(w0 * N[(1.0 + N[(t$95$0 * N[(N[(D$95$m * N[(h * N[(D$95$m * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1.6e+36], w0, N[(w0 * N[(1.0 + N[(t$95$0 * N[(N[(D$95$m * N[(D$95$m * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.9000000000000003e-130 or 2.2000000000000001e-21 < M < 1.5999999999999999e36

   1. Initial program 82.1%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 70.8%

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

   if 5.9000000000000003e-130 < M < 2.2000000000000001e-21

   1. Initial program 80.4%

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

   3. Simplified 93.2%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. /-lowering-/.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in h around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. distribute-rgt-in N/A

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

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

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

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

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

      12. associate-*r/ N/A

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

   9. Simplified 86.1%

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

   if 1.5999999999999999e36 < M

   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 79.9%

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

   5. Taylor expanded in h around 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 43.5%

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

   7. Simplified 43.5%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.9 \cdot 10^{-130}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-21}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{d \cdot d} \cdot \frac{D \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;M \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{d \cdot d} \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := w0 \cdot \left(1 + \frac{-0.125}{d\_m \cdot d\_m} \cdot \frac{D\_m \cdot \left(D\_m \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \mathbf{if}\;M \leq 5.2 \cdot 10^{-154}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (let* ((t_0
         (*
          w0
          (+
           1.0
           (* (/ -0.125 (* d_m d_m)) (/ (* D_m (* D_m (* h (* M M)))) l))))))
   (if (<= M 5.2e-154) w0 (if (<= M 2.6e-23) t_0 (if (<= M 2.2e+36) w0 t_0)))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = w0 * (1.0 + ((-0.125 / (d_m * d_m)) * ((D_m * (D_m * (h * (M * M)))) / l)));
	double tmp;
	if (M <= 5.2e-154) {
		tmp = w0;
	} else if (M <= 2.6e-23) {
		tmp = t_0;
	} else if (M <= 2.2e+36) {
		tmp = w0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 = w0 * (1.0d0 + (((-0.125d0) / (d_m_1 * d_m_1)) * ((d_m * (d_m * (h * (m * m)))) / l)))
    if (m <= 5.2d-154) then
        tmp = w0
    else if (m <= 2.6d-23) then
        tmp = t_0
    else if (m <= 2.2d+36) then
        tmp = w0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double t_0 = w0 * (1.0 + ((-0.125 / (d_m * d_m)) * ((D_m * (D_m * (h * (M * M)))) / l)));
	double tmp;
	if (M <= 5.2e-154) {
		tmp = w0;
	} else if (M <= 2.6e-23) {
		tmp = t_0;
	} else if (M <= 2.2e+36) {
		tmp = w0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	t_0 = w0 * (1.0 + ((-0.125 / (d_m * d_m)) * ((D_m * (D_m * (h * (M * M)))) / l)))
	tmp = 0
	if M <= 5.2e-154:
		tmp = w0
	elif M <= 2.6e-23:
		tmp = t_0
	elif M <= 2.2e+36:
		tmp = w0
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	t_0 = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / Float64(d_m * d_m)) * Float64(Float64(D_m * Float64(D_m * Float64(h * Float64(M * M)))) / l))))
	tmp = 0.0
	if (M <= 5.2e-154)
		tmp = w0;
	elseif (M <= 2.6e-23)
		tmp = t_0;
	elseif (M <= 2.2e+36)
		tmp = w0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	t_0 = w0 * (1.0 + ((-0.125 / (d_m * d_m)) * ((D_m * (D_m * (h * (M * M)))) / l)));
	tmp = 0.0;
	if (M <= 5.2e-154)
		tmp = w0;
	elseif (M <= 2.6e-23)
		tmp = t_0;
	elseif (M <= 2.2e+36)
		tmp = w0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(w0 * N[(1.0 + N[(N[(-0.125 / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(D$95$m * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 5.2e-154], w0, If[LessEqual[M, 2.6e-23], t$95$0, If[LessEqual[M, 2.2e+36], w0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if M < 5.2e-154 or 2.6e-23 < M < 2.2e36

   1. Initial program 82.2%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 70.8%

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

   if 5.2e-154 < M < 2.6e-23 or 2.2e36 < M

   1. Initial program 83.5%

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

   3. Simplified 83.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   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 52.9%

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

   7. Simplified 52.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.2 \cdot 10^{-154}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{d \cdot d} \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{d \cdot d} \cdot \frac{D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;w0 + \left(D\_m \cdot -0.125\right) \cdot \frac{\frac{\left(M \cdot \frac{M \cdot D\_m}{d\_m}\right) \cdot \left(h \cdot w0\right)}{\ell}}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(0.125 \cdot \left(D\_m \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot \frac{D\_m}{\left(d\_m \cdot \ell\right) \cdot \left(0 - d\_m\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= d_m 1.6e-6)
   (+ w0 (* (* D_m -0.125) (/ (/ (* (* M (/ (* M D_m) d_m)) (* h w0)) l) d_m)))
   (*
    w0
    (-
     (* (* 0.125 (* D_m (* M (* M h)))) (/ D_m (* (* d_m l) (- 0.0 d_m))))
     -1.0))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.6e-6) {
		tmp = w0 + ((D_m * -0.125) * ((((M * ((M * D_m) / d_m)) * (h * w0)) / l) / d_m));
	} else {
		tmp = w0 * (((0.125 * (D_m * (M * (M * h)))) * (D_m / ((d_m * l) * (0.0 - d_m)))) - -1.0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 1.6d-6) then
        tmp = w0 + ((d_m * (-0.125d0)) * ((((m * ((m * d_m) / d_m_1)) * (h * w0)) / l) / d_m_1))
    else
        tmp = w0 * (((0.125d0 * (d_m * (m * (m * h)))) * (d_m / ((d_m_1 * l) * (0.0d0 - d_m_1)))) - (-1.0d0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.6e-6) {
		tmp = w0 + ((D_m * -0.125) * ((((M * ((M * D_m) / d_m)) * (h * w0)) / l) / d_m));
	} else {
		tmp = w0 * (((0.125 * (D_m * (M * (M * h)))) * (D_m / ((d_m * l) * (0.0 - d_m)))) - -1.0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 1.6e-6:
		tmp = w0 + ((D_m * -0.125) * ((((M * ((M * D_m) / d_m)) * (h * w0)) / l) / d_m))
	else:
		tmp = w0 * (((0.125 * (D_m * (M * (M * h)))) * (D_m / ((d_m * l) * (0.0 - d_m)))) - -1.0)
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 1.6e-6)
		tmp = Float64(w0 + Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(Float64(M * Float64(Float64(M * D_m) / d_m)) * Float64(h * w0)) / l) / d_m)));
	else
		tmp = Float64(w0 * Float64(Float64(Float64(0.125 * Float64(D_m * Float64(M * Float64(M * h)))) * Float64(D_m / Float64(Float64(d_m * l) * Float64(0.0 - d_m)))) - -1.0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 1.6e-6)
		tmp = w0 + ((D_m * -0.125) * ((((M * ((M * D_m) / d_m)) * (h * w0)) / l) / d_m));
	else
		tmp = w0 * (((0.125 * (D_m * (M * (M * h)))) * (D_m / ((d_m * l) * (0.0 - d_m)))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 1.6e-6], N[(w0 + N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(N[(M * N[(N[(M * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(0.125 * N[(D$95$m * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(N[(d$95$m * l), $MachinePrecision] * N[(0.0 - d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.5999999999999999e-6

   1. Initial program 81.9%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 47.0%

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

   7. Simplified 47.0%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. *-lowering-*.f64 67.7%

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

   11. Applied egg-rr 67.7%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

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

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

   13. Applied egg-rr 74.6%

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

   if 1.5999999999999999e-6 < d

   1. Initial program 84.7%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in w0 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

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

   10. Simplified 64.5%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. associate-*l* N/A

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

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

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

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

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

       13. associate-*r* N/A

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

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

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

       15. *-lowering-*.f64 66.2%

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

   12. Applied egg-rr 66.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \frac{\frac{\left(M \cdot \frac{M \cdot D}{d}\right) \cdot \left(h \cdot w0\right)}{\ell}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(0.125 \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot \frac{D}{\left(d \cdot \ell\right) \cdot \left(0 - d\right)} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 3.6 \cdot 10^{+58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D\_m \cdot -0.125\right) \cdot \left(\left(D\_m \cdot \frac{M}{d\_m}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot w0}{d\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= D_m 3.6e+58)
   w0
   (+
    w0
    (* (* D_m -0.125) (* (* D_m (/ M d_m)) (* (/ h l) (/ (* M w0) d_m)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 3.6e+58) {
		tmp = w0;
	} else {
		tmp = w0 + ((D_m * -0.125) * ((D_m * (M / d_m)) * ((h / l) * ((M * w0) / d_m))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 3.6d+58) then
        tmp = w0
    else
        tmp = w0 + ((d_m * (-0.125d0)) * ((d_m * (m / d_m_1)) * ((h / l) * ((m * w0) / d_m_1))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 3.6e+58) {
		tmp = w0;
	} else {
		tmp = w0 + ((D_m * -0.125) * ((D_m * (M / d_m)) * ((h / l) * ((M * w0) / d_m))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 3.6e+58:
		tmp = w0
	else:
		tmp = w0 + ((D_m * -0.125) * ((D_m * (M / d_m)) * ((h / l) * ((M * w0) / d_m))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 3.6e+58)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(Float64(D_m * -0.125) * Float64(Float64(D_m * Float64(M / d_m)) * Float64(Float64(h / l) * Float64(Float64(M * w0) / d_m)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 3.6e+58)
		tmp = w0;
	else
		tmp = w0 + ((D_m * -0.125) * ((D_m * (M / d_m)) * ((h / l) * ((M * w0) / d_m))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 3.6e+58], w0, N[(w0 + N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(D$95$m * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(M * w0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.59999999999999996e58

   1. Initial program 81.0%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 73.3%

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

   if 3.59999999999999996e58 < D

   1. Initial program 90.6%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 43.3%

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

   7. Simplified 43.3%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. *-lowering-*.f64 71.0%

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

   11. Applied egg-rr 71.0%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

       7. /-lowering-/.f64 78.2%

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

   13. Applied egg-rr 78.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.6 \cdot 10^{+58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot w0}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.65 \cdot 10^{+89}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D\_m \cdot -0.125\right) \cdot \left(D\_m \cdot \frac{M \cdot \left(w0 \cdot \left(M \cdot h\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= D_m 1.65e+89)
   w0
   (+
    w0
    (* (* D_m -0.125) (* D_m (/ (* M (* w0 (* M h))) (* d_m (* d_m l))))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.65e+89) {
		tmp = w0;
	} else {
		tmp = w0 + ((D_m * -0.125) * (D_m * ((M * (w0 * (M * h))) / (d_m * (d_m * l)))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 1.65d+89) then
        tmp = w0
    else
        tmp = w0 + ((d_m * (-0.125d0)) * (d_m * ((m * (w0 * (m * h))) / (d_m_1 * (d_m_1 * l)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.65e+89) {
		tmp = w0;
	} else {
		tmp = w0 + ((D_m * -0.125) * (D_m * ((M * (w0 * (M * h))) / (d_m * (d_m * l)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 1.65e+89:
		tmp = w0
	else:
		tmp = w0 + ((D_m * -0.125) * (D_m * ((M * (w0 * (M * h))) / (d_m * (d_m * l)))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.65e+89)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(Float64(D_m * -0.125) * Float64(D_m * Float64(Float64(M * Float64(w0 * Float64(M * h))) / Float64(d_m * Float64(d_m * l))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 1.65e+89)
		tmp = w0;
	else
		tmp = w0 + ((D_m * -0.125) * (D_m * ((M * (w0 * (M * h))) / (d_m * (d_m * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 1.65e+89], w0, N[(w0 + N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(D$95$m * N[(N[(M * N[(w0 * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.64999999999999987e89

   1. Initial program 81.6%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 73.3%

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

   if 1.64999999999999987e89 < D

   1. Initial program 88.7%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 67.8%

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

   9. Applied egg-rr 67.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.65 \cdot 10^{+89}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \left(D \cdot \frac{M \cdot \left(w0 \cdot \left(M \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d\_m}}{\ell} \cdot \left(D\_m \cdot \frac{D\_m \cdot -0.125}{d\_m}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 1.1e+36)
   w0
   (* (/ (/ (* M (* M (* h w0))) d_m) l) (* D_m (/ (* D_m -0.125) d_m)))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 1.1e+36) {
		tmp = w0;
	} else {
		tmp = (((M * (M * (h * w0))) / d_m) / l) * (D_m * ((D_m * -0.125) / d_m));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 1.1d+36) then
        tmp = w0
    else
        tmp = (((m * (m * (h * w0))) / d_m_1) / l) * (d_m * ((d_m * (-0.125d0)) / d_m_1))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 1.1e+36) {
		tmp = w0;
	} else {
		tmp = (((M * (M * (h * w0))) / d_m) / l) * (D_m * ((D_m * -0.125) / d_m));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 1.1e+36:
		tmp = w0
	else:
		tmp = (((M * (M * (h * w0))) / d_m) / l) * (D_m * ((D_m * -0.125) / d_m))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 1.1e+36)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(Float64(M * Float64(M * Float64(h * w0))) / d_m) / l) * Float64(D_m * Float64(Float64(D_m * -0.125) / d_m)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 1.1e+36)
		tmp = w0;
	else
		tmp = (((M * (M * (h * w0))) / d_m) / l) * (D_m * ((D_m * -0.125) / d_m));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 1.1e+36], w0, N[(N[(N[(N[(M * N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.1e36

   1. Initial program 82.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.1%

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

   if 1.1e36 < M

   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 79.9%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in D around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

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

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 19.2%

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

   10. Simplified 19.2%

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. associate-/r* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. associate-*l/ N/A

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

       12. *-commutative N/A

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

       13. times-frac N/A

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

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

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

   12. Applied egg-rr 25.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d}}{\ell} \cdot \left(D \cdot \frac{D \cdot -0.125}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \frac{-0.125 \cdot \left(\left(M \cdot \frac{M \cdot D\_m}{d\_m}\right) \cdot \left(h \cdot w0\right)\right)}{d\_m \cdot \ell}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 3.7e+36)
   w0
   (* D_m (/ (* -0.125 (* (* M (/ (* M D_m) d_m)) (* h w0))) (* d_m l)))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.7e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((-0.125 * ((M * ((M * D_m) / d_m)) * (h * w0))) / (d_m * l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 3.7d+36) then
        tmp = w0
    else
        tmp = d_m * (((-0.125d0) * ((m * ((m * d_m) / d_m_1)) * (h * w0))) / (d_m_1 * l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.7e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((-0.125 * ((M * ((M * D_m) / d_m)) * (h * w0))) / (d_m * l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 3.7e+36:
		tmp = w0
	else:
		tmp = D_m * ((-0.125 * ((M * ((M * D_m) / d_m)) * (h * w0))) / (d_m * l))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 3.7e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(M * Float64(Float64(M * D_m) / d_m)) * Float64(h * w0))) / Float64(d_m * l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 3.7e+36)
		tmp = w0;
	else
		tmp = D_m * ((-0.125 * ((M * ((M * D_m) / d_m)) * (h * w0))) / (d_m * l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 3.7e+36], w0, N[(D$95$m * N[(N[(-0.125 * N[(N[(M * N[(N[(M * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.70000000000000029e36

   1. Initial program 82.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.1%

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

   if 3.70000000000000029e36 < M

   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 79.9%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in D around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

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

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

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

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

      12. associate-*r* N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 19.2%

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

   10. Simplified 19.2%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

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

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

   12. Applied egg-rr 25.9%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. times-frac N/A

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

       5. associate-*l* N/A

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

       6. associate-*r/ N/A

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

       7. associate-*r/ N/A

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

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

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

   14. Applied egg-rr 27.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \frac{-0.125 \cdot \left(\left(M \cdot \frac{M \cdot D}{d}\right) \cdot \left(h \cdot w0\right)\right)}{d \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(\frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d\_m} \cdot \frac{\frac{D\_m \cdot -0.125}{\ell}}{d\_m}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 4.2e+36)
   w0
   (* D_m (* (/ (* M (* M (* h w0))) d_m) (/ (/ (* D_m -0.125) l) d_m)))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.2e+36) {
		tmp = w0;
	} else {
		tmp = D_m * (((M * (M * (h * w0))) / d_m) * (((D_m * -0.125) / l) / d_m));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 4.2d+36) then
        tmp = w0
    else
        tmp = d_m * (((m * (m * (h * w0))) / d_m_1) * (((d_m * (-0.125d0)) / l) / d_m_1))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.2e+36) {
		tmp = w0;
	} else {
		tmp = D_m * (((M * (M * (h * w0))) / d_m) * (((D_m * -0.125) / l) / d_m));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 4.2e+36:
		tmp = w0
	else:
		tmp = D_m * (((M * (M * (h * w0))) / d_m) * (((D_m * -0.125) / l) / d_m))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 4.2e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(Float64(M * Float64(M * Float64(h * w0))) / d_m) * Float64(Float64(Float64(D_m * -0.125) / l) / d_m)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 4.2e+36)
		tmp = w0;
	else
		tmp = D_m * (((M * (M * (h * w0))) / d_m) * (((D_m * -0.125) / l) / d_m));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 4.2e+36], w0, N[(D$95$m * N[(N[(N[(M * N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(N[(D$95$m * -0.125), $MachinePrecision] / l), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.20000000000000009e36

   1. Initial program 82.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 71.1%

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

   if 4.20000000000000009e36 < M

   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 79.9%

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

   5. Taylor expanded in h around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{w0 + \frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \frac{-1}{8}\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(h \cdot M\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(\color{blue}{d} \cdot d\right) \cdot \ell} \]

       8. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(d \cdot \ell\right) \cdot \color{blue}{d}} \]

       10. frac-times N/A

           \[\leadsto \frac{D \cdot \left(D \cdot \frac{-1}{8}\right)}{d \cdot \ell} \cdot \color{blue}{\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d}} \]

       11. associate-/l* N/A

           \[\leadsto \left(D \cdot \frac{D \cdot \frac{-1}{8}}{d \cdot \ell}\right) \cdot \frac{\color{blue}{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}}{d} \]

       12. associate-*l* N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{D \cdot \frac{-1}{8}}{d \cdot \ell} \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d}\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{D \cdot \frac{-1}{8}}{d \cdot \ell} \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d}\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{D \cdot \frac{-1}{8}}{d \cdot \ell}\right), \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d}\right)}\right)\right) \]

   12. Applied egg-rr 25.7%

       \[\leadsto \color{blue}{D \cdot \left(\frac{\frac{D \cdot -0.125}{\ell}}{d} \cdot \frac{M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{d}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(\frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d} \cdot \frac{\frac{D \cdot -0.125}{\ell}}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \frac{\frac{M}{d\_m} \cdot \frac{M \cdot \left(h \cdot w0\right)}{d\_m}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 2.9e+36)
   w0
   (* D_m (* (* D_m -0.125) (/ (* (/ M d_m) (/ (* M (* h w0)) d_m)) l)))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 2.9e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * ((M * (h * w0)) / d_m)) / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 2.9d+36) then
        tmp = w0
    else
        tmp = d_m * ((d_m * (-0.125d0)) * (((m / d_m_1) * ((m * (h * w0)) / d_m_1)) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 2.9e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * ((M * (h * w0)) / d_m)) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 2.9e+36:
		tmp = w0
	else:
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * ((M * (h * w0)) / d_m)) / l))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 2.9e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M / d_m) * Float64(Float64(M * Float64(h * w0)) / d_m)) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 2.9e+36)
		tmp = w0;
	else
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * ((M * (h * w0)) / d_m)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 2.9e+36], w0, N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M / d$95$m), $MachinePrecision] * N[(N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9e36

   1. Initial program 82.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.1%

      \[\leadsto \color{blue}{w0} \]

   if 2.9e36 < M

   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 79.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{w0 + \frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \frac{-1}{8}\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(h \cdot M\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(\color{blue}{d} \cdot d\right) \cdot \ell} \]

       8. associate-*r/ N/A

          \[\leadsto \left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}} \]

       9. *-commutative N/A

          \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}}{\left(d \cdot d\right) \cdot \ell} \]

       10. associate-*r* N/A

           \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       11. associate-*r* N/A

           \[\leadsto \left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \color{blue}{\left(D \cdot \frac{-1}{8}\right)} \]

       13. associate-*l* N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)}\right) \]

   12. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{D \cdot \left(\frac{M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot -0.125\right)\right)} \]
   13. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{M}{d} \cdot \frac{M \cdot \left(w0 \cdot h\right)}{d \cdot \ell}\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{M}{d} \cdot \frac{\frac{M \cdot \left(w0 \cdot h\right)}{d}}{\ell}\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{\frac{M}{d} \cdot \frac{M \cdot \left(w0 \cdot h\right)}{d}}{\ell}\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{d} \cdot \frac{M \cdot \left(w0 \cdot h\right)}{d}\right), \ell\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{M}{d}\right), \left(\frac{M \cdot \left(w0 \cdot h\right)}{d}\right)\right), \ell\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \left(\frac{M \cdot \left(w0 \cdot h\right)}{d}\right)\right), \ell\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(\left(M \cdot \left(w0 \cdot h\right)\right), d\right)\right), \ell\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(w0 \cdot h\right)\right), d\right)\right), \ell\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       9. *-lowering-*.f64 28.1%

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(w0, h\right)\right), d\right)\right), \ell\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

   14. Applied egg-rr 28.1%

       \[\leadsto D \cdot \left(\color{blue}{\frac{\frac{M}{d} \cdot \frac{M \cdot \left(w0 \cdot h\right)}{d}}{\ell}} \cdot \left(D \cdot -0.125\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(\left(D \cdot -0.125\right) \cdot \frac{\frac{M}{d} \cdot \frac{M \cdot \left(h \cdot w0\right)}{d}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \frac{\left(M \cdot w0\right) \cdot \left(M \cdot h\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 3.5e+36)
   w0
   (* D_m (* (* D_m -0.125) (/ (* (* M w0) (* M h)) (* d_m (* d_m l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.5e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M * w0) * (M * h)) / (d_m * (d_m * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 3.5d+36) then
        tmp = w0
    else
        tmp = d_m * ((d_m * (-0.125d0)) * (((m * w0) * (m * h)) / (d_m_1 * (d_m_1 * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.5e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M * w0) * (M * h)) / (d_m * (d_m * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 3.5e+36:
		tmp = w0
	else:
		tmp = D_m * ((D_m * -0.125) * (((M * w0) * (M * h)) / (d_m * (d_m * l))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 3.5e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M * w0) * Float64(M * h)) / Float64(d_m * Float64(d_m * l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 3.5e+36)
		tmp = w0;
	else
		tmp = D_m * ((D_m * -0.125) * (((M * w0) * (M * h)) / (d_m * (d_m * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 3.5e+36], w0, N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M * w0), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.4999999999999998e36

   1. Initial program 82.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.1%

      \[\leadsto \color{blue}{w0} \]

   if 3.4999999999999998e36 < M

   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 79.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{w0 + \frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \frac{-1}{8}\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(h \cdot M\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(\color{blue}{d} \cdot d\right) \cdot \ell} \]

       8. associate-*r/ N/A

          \[\leadsto \left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}} \]

       9. *-commutative N/A

          \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}}{\left(d \cdot d\right) \cdot \ell} \]

       10. associate-*r* N/A

           \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       11. associate-*r* N/A

           \[\leadsto \left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \color{blue}{\left(D \cdot \frac{-1}{8}\right)} \]

       13. associate-*l* N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)}\right) \]

   12. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{D \cdot \left(\frac{M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot -0.125\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(M \cdot \left(w0 \cdot h\right)\right) \cdot M\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(M \cdot w0\right) \cdot h\right) \cdot M\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(M \cdot w0\right) \cdot \left(h \cdot M\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot w0\right), \left(h \cdot M\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(w0 \cdot M\right), \left(h \cdot M\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w0, M\right), \left(h \cdot M\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       7. *-lowering-*.f64 25.7%

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w0, M\right), \mathsf{*.f64}\left(h, M\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \ell\right)\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

   14. Applied egg-rr 25.7%

       \[\leadsto D \cdot \left(\frac{\color{blue}{\left(w0 \cdot M\right) \cdot \left(h \cdot M\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot -0.125\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.5 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(\left(D \cdot -0.125\right) \cdot \frac{\left(M \cdot w0\right) \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 4.2e+36)
   w0
   (* D_m (* (* D_m -0.125) (/ (* M (* M (* h w0))) (* d_m (* d_m l)))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.2e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * ((M * (M * (h * w0))) / (d_m * (d_m * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 4.2d+36) then
        tmp = w0
    else
        tmp = d_m * ((d_m * (-0.125d0)) * ((m * (m * (h * w0))) / (d_m_1 * (d_m_1 * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 4.2e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * ((M * (M * (h * w0))) / (d_m * (d_m * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 4.2e+36:
		tmp = w0
	else:
		tmp = D_m * ((D_m * -0.125) * ((M * (M * (h * w0))) / (d_m * (d_m * l))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 4.2e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(D_m * -0.125) * Float64(Float64(M * Float64(M * Float64(h * w0))) / Float64(d_m * Float64(d_m * l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 4.2e+36)
		tmp = w0;
	else
		tmp = D_m * ((D_m * -0.125) * ((M * (M * (h * w0))) / (d_m * (d_m * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 4.2e+36], w0, N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(M * N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.20000000000000009e36

   1. Initial program 82.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.1%

      \[\leadsto \color{blue}{w0} \]

   if 4.20000000000000009e36 < M

   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 79.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{w0 + \frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \frac{-1}{8}\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(h \cdot M\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(\color{blue}{d} \cdot d\right) \cdot \ell} \]

       8. associate-*r/ N/A

          \[\leadsto \left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}} \]

       9. *-commutative N/A

          \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}}{\left(d \cdot d\right) \cdot \ell} \]

       10. associate-*r* N/A

           \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       11. associate-*r* N/A

           \[\leadsto \left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \color{blue}{\left(D \cdot \frac{-1}{8}\right)} \]

       13. associate-*l* N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)}\right) \]

   12. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{D \cdot \left(\frac{M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot -0.125\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(\left(D \cdot -0.125\right) \cdot \frac{M \cdot \left(M \cdot \left(h \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D\_m \cdot \left(\left(D\_m \cdot -0.125\right) \cdot \left(\left(\frac{M}{d\_m} \cdot \frac{M}{d\_m}\right) \cdot \frac{h \cdot w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m)
 :precision binary64
 (if (<= M 3.8e+36)
   w0
   (* D_m (* (* D_m -0.125) (* (* (/ M d_m) (/ M d_m)) (/ (* h w0) l))))))

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.8e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * (M / d_m)) * ((h * w0) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 <= 3.8d+36) then
        tmp = w0
    else
        tmp = d_m * ((d_m * (-0.125d0)) * (((m / d_m_1) * (m / d_m_1)) * ((h * w0) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M <= 3.8e+36) {
		tmp = w0;
	} else {
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * (M / d_m)) * ((h * w0) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	tmp = 0
	if M <= 3.8e+36:
		tmp = w0
	else:
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * (M / d_m)) * ((h * w0) / l)))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	tmp = 0.0
	if (M <= 3.8e+36)
		tmp = w0;
	else
		tmp = Float64(D_m * Float64(Float64(D_m * -0.125) * Float64(Float64(Float64(M / d_m) * Float64(M / d_m)) * Float64(Float64(h * w0) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M, D_m, h, l, d_m)
	tmp = 0.0;
	if (M <= 3.8e+36)
		tmp = w0;
	else
		tmp = D_m * ((D_m * -0.125) * (((M / d_m) * (M / d_m)) * ((h * w0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M, 3.8e+36], w0, N[(D$95$m * N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(N[(M / d$95$m), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.80000000000000025e36

   1. Initial program 82.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.1%

      \[\leadsto \color{blue}{w0} \]

   if 3.80000000000000025e36 < M

   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 79.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 37.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   7. Simplified 37.9%

      \[\leadsto \color{blue}{w0 + \frac{\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   8. Taylor expanded in D around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left({D}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot \frac{-1}{8}\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \frac{-1}{8}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), w0\right), \frac{-1}{8}\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      19. *-lowering-*.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), w0\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot -0.125}{\left(d \cdot d\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \frac{-1}{8}\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right)\right)}{\left(d \cdot \color{blue}{d}\right) \cdot \ell} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(h \cdot M\right)\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot D\right) \cdot \left(\frac{-1}{8} \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       7. associate-*r* N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot w0\right)\right)}{\left(\color{blue}{d} \cdot d\right) \cdot \ell} \]

       8. associate-*r/ N/A

          \[\leadsto \left(D \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{\left(d \cdot d\right) \cdot \ell}} \]

       9. *-commutative N/A

          \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{\color{blue}{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}}{\left(d \cdot d\right) \cdot \ell} \]

       10. associate-*r* N/A

           \[\leadsto \left(\left(D \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       11. associate-*r* N/A

           \[\leadsto \left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(D \cdot \frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \color{blue}{\left(D \cdot \frac{-1}{8}\right)} \]

       13. associate-*l* N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot w0\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right)}\right) \]

   12. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{D \cdot \left(\frac{M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(D \cdot -0.125\right)\right)} \]
   13. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{M}{d} \cdot \frac{M \cdot \left(w0 \cdot h\right)}{d \cdot \ell}\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot \frac{w0 \cdot h}{\ell}\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \frac{w0 \cdot h}{\ell}\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right), \left(\frac{w0 \cdot h}{\ell}\right)\right), \mathsf{*.f64}\left(\color{blue}{D}, \frac{-1}{8}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M}{d}\right), \left(\frac{M}{d}\right)\right), \left(\frac{w0 \cdot h}{\ell}\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \left(\frac{M}{d}\right)\right), \left(\frac{w0 \cdot h}{\ell}\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(M, d\right)\right), \left(\frac{w0 \cdot h}{\ell}\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(M, d\right)\right), \mathsf{/.f64}\left(\left(w0 \cdot h\right), \ell\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

       9. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(M, d\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, h\right), \ell\right)\right), \mathsf{*.f64}\left(D, \frac{-1}{8}\right)\right)\right) \]

   14. Applied egg-rr 27.6%

       \[\leadsto D \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \frac{w0 \cdot h}{\ell}\right)} \cdot \left(D \cdot -0.125\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(\left(D \cdot -0.125\right) \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \frac{h \cdot w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.2% accurate, 216.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M, D_m, h, l, d_m] = \mathsf{sort}([w0, M, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d_m) :precision binary64 w0)

C

D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

D_m = abs(d)
d_m = abs(d)
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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

D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M && M < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M, double D_m, double h, double l, double d_m) {
	return w0;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M, D_m, h, l, d_m] = sort([w0, M, D_m, h, l, d_m])
def code(w0, M, D_m, h, l, d_m):
	return w0

Julia

D_m = abs(D)
d_m = abs(d)
w0, M, D_m, h, l, d_m = sort([w0, M, D_m, h, l, d_m])
function code(w0, M, D_m, h, l, d_m)
	return w0
end

MATLAB

D_m = abs(D);
d_m = abs(d);
w0, M, D_m, h, l, d_m = num2cell(sort([w0, M, D_m, h, l, d_m])){:}
function tmp = code(w0, M, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 82.6%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 84.8%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h \cdot \left(D \cdot \frac{M \cdot \frac{M \cdot D}{d}}{-4}\right)}{d}}{\ell}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 68.4%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))))