Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 83.6%

Time: 13.1s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% 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: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \cdot M\_m \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{\frac{\frac{D\_m \cdot \left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right)}{d}}{d \cdot 4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M\_m \cdot \frac{\left(M\_m \cdot 0.5\right) \cdot \left(D\_m \cdot h\right)}{d \cdot \ell}, \frac{D\_m}{d \cdot -2}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* D_m M_m) 5e+143)
   (*
    w0
    (sqrt (- 1.0 (* h (/ (/ (/ (* D_m (* M_m (* D_m M_m))) d) (* d 4.0)) l)))))
   (*
    w0
    (sqrt
     (fma
      (* M_m (/ (* (* M_m 0.5) (* D_m h)) (* d l)))
      (/ D_m (* d -2.0))
      1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((D_m * M_m) <= 5e+143) {
		tmp = w0 * sqrt((1.0 - (h * ((((D_m * (M_m * (D_m * M_m))) / d) / (d * 4.0)) / l))));
	} else {
		tmp = w0 * sqrt(fma((M_m * (((M_m * 0.5) * (D_m * h)) / (d * l))), (D_m / (d * -2.0)), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(D_m * M_m) <= 5e+143)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(Float64(D_m * Float64(M_m * Float64(D_m * M_m))) / d) / Float64(d * 4.0)) / l)))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(M_m * Float64(Float64(Float64(M_m * 0.5) * Float64(D_m * h)) / Float64(d * l))), Float64(D_m / Float64(d * -2.0)), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(D$95$m * M$95$m), $MachinePrecision], 5e+143], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(N[(D$95$m * N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.00000000000000012e143

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied egg-rr 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 86.0

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

   6. Applied egg-rr 86.0%

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

   if 5.00000000000000012e143 < (*.f64 M D)

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. *-commutative N/A

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

   4. Applied egg-rr 80.7%

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

   6. Applied egg-rr 80.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{\frac{\frac{D \cdot \left(M \cdot \left(D \cdot M\right)\right)}{d}}{d \cdot 4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M \cdot \frac{\left(M \cdot 0.5\right) \cdot \left(D \cdot h\right)}{d \cdot \ell}, \frac{D}{d \cdot -2}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M\_m \cdot 0.5\right) \cdot \left(D\_m \cdot h\right)}{d \cdot \ell} \cdot \frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -0.01)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ (* (* M_m 0.5) (* D_m h)) (* d l)) (/ (* M_m (* D_m 0.5)) d)))))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * sqrt((1.0 - ((((M_m * 0.5) * (D_m * h)) / (d * l)) * ((M_m * (D_m * 0.5)) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-0.01d0)) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * 0.5d0) * (d_m * h)) / (d * l)) * ((m_m * (d_m * 0.5d0)) / d))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * 0.5) * (D_m * h)) / (d * l)) * ((M_m * (D_m * 0.5)) / d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * 0.5) * (D_m * h)) / (d * l)) * ((M_m * (D_m * 0.5)) / d))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * 0.5) * Float64(D_m * h)) / Float64(d * l)) * Float64(Float64(M_m * Float64(D_m * 0.5)) / d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.01)
		tmp = w0 * sqrt((1.0 - ((((M_m * 0.5) * (D_m * h)) / (d * l)) * ((M_m * (D_m * 0.5)) / d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. *-commutative N/A

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

   4. Applied egg-rr 66.1%

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

   6. Applied egg-rr 68.3%

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

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot 0.5\right) \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{M \cdot \left(D \cdot 0.5\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M\_m \cdot \frac{\left(M\_m \cdot 0.5\right) \cdot \left(D\_m \cdot h\right)}{d \cdot \ell}, \frac{D\_m}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -0.01)
   (*
    w0
    (sqrt
     (fma
      (* M_m (/ (* (* M_m 0.5) (* D_m h)) (* d l)))
      (/ D_m (* d -2.0))
      1.0)))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * sqrt(fma((M_m * (((M_m * 0.5) * (D_m * h)) / (d * l))), (D_m / (d * -2.0)), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.01)
		tmp = Float64(w0 * sqrt(fma(Float64(M_m * Float64(Float64(Float64(M_m * 0.5) * Float64(D_m * h)) / Float64(d * l))), Float64(D_m / Float64(d * -2.0)), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.01], N[(w0 * N[Sqrt[N[(N[(M$95$m * N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. *-commutative N/A

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

   4. Applied egg-rr 66.1%

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

   6. Applied egg-rr 62.7%

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

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M \cdot \frac{\left(M \cdot 0.5\right) \cdot \left(D \cdot h\right)}{d \cdot \ell}, \frac{D}{d \cdot -2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D\_m \cdot M\_m\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(\left(D\_m \cdot M\_m\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -0.01)
   (*
    w0
    (sqrt
     (- 1.0 (* (* (* D_m M_m) (/ h (* d (* d l)))) (* (* D_m M_m) 0.25)))))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * sqrt((1.0 - (((D_m * M_m) * (h / (d * (d * l)))) * ((D_m * M_m) * 0.25))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-0.01d0)) then
        tmp = w0 * sqrt((1.0d0 - (((d_m * m_m) * (h / (d * (d * l)))) * ((d_m * m_m) * 0.25d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * Math.sqrt((1.0 - (((D_m * M_m) * (h / (d * (d * l)))) * ((D_m * M_m) * 0.25))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01:
		tmp = w0 * math.sqrt((1.0 - (((D_m * M_m) * (h / (d * (d * l)))) * ((D_m * M_m) * 0.25))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * M_m) * Float64(h / Float64(d * Float64(d * l)))) * Float64(Float64(D_m * M_m) * 0.25)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.01)
		tmp = w0 * sqrt((1.0 - (((D_m * M_m) * (h / (d * (d * l)))) * ((D_m * M_m) * 0.25))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(h / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.9

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

   4. Applied egg-rr 60.9%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   6. Applied egg-rr 67.7%

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

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D \cdot M\right) \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(\left(D \cdot M\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m \cdot M\_m}{4 \cdot \left(d \cdot \left(d \cdot \ell\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -0.01)
   (*
    w0
    (sqrt (- 1.0 (* h (* (* D_m M_m) (/ (* D_m M_m) (* 4.0 (* d (* d l)))))))))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * sqrt((1.0 - (h * ((D_m * M_m) * ((D_m * M_m) / (4.0 * (d * (d * l))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-0.01d0)) then
        tmp = w0 * sqrt((1.0d0 - (h * ((d_m * m_m) * ((d_m * m_m) / (4.0d0 * (d * (d * l))))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((D_m * M_m) * ((D_m * M_m) / (4.0 * (d * (d * l))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.01:
		tmp = w0 * math.sqrt((1.0 - (h * ((D_m * M_m) * ((D_m * M_m) / (4.0 * (d * (d * l))))))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(D_m * M_m) * Float64(Float64(D_m * M_m) / Float64(4.0 * Float64(d * Float64(d * l)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -0.01)
		tmp = w0 * sqrt((1.0 - (h * ((D_m * M_m) * ((D_m * M_m) / (4.0 * (d * (d * l))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(4.0 * N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied egg-rr 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

   6. Applied egg-rr 66.3%

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

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.4%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{4 \cdot \left(d \cdot \left(d \cdot \ell\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \left(h \cdot \left(D\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0 \cdot -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (fma (/ (* D_m (* M_m (* h (* D_m M_m)))) (* d (* d l))) (* w0 -0.125) w0)
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = fma(((D_m * (M_m * (h * (D_m * M_m)))) / (d * (d * l))), (w0 * -0.125), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = fma(Float64(Float64(D_m * Float64(M_m * Float64(h * Float64(D_m * M_m)))) / Float64(d * Float64(d * l))), Float64(w0 * -0.125), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(N[(D$95$m * N[(M$95$m * N[(h * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 * -0.125), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 56.7%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 61.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.5

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

   10. Applied egg-rr 63.5%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0 \cdot -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D\_m \cdot \left(h \cdot \left(M\_m \cdot \left(D\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0 \cdot -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (fma (/ (* D_m (* h (* M_m (* D_m M_m)))) (* d (* d l))) (* w0 -0.125) w0)
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = fma(((D_m * (h * (M_m * (D_m * M_m)))) / (d * (d * l))), (w0 * -0.125), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = fma(Float64(Float64(D_m * Float64(h * Float64(M_m * Float64(D_m * M_m)))) / Float64(d * Float64(d * l))), Float64(w0 * -0.125), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(N[(D$95$m * N[(h * N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 * -0.125), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 56.7%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 61.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.5

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

   10. Applied egg-rr 63.5%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D \cdot \left(h \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0 \cdot -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot \left(\left(h \cdot \left(M\_m \cdot \left(w0 \cdot M\_m\right)\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right), D\_m, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (fma (* D_m (* (* h (* M_m (* w0 M_m))) (/ -0.125 (* d (* d l))))) D_m w0)
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = fma((D_m * ((h * (M_m * (w0 * M_m))) * (-0.125 / (d * (d * l))))), D_m, w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = fma(Float64(D_m * Float64(Float64(h * Float64(M_m * Float64(w0 * M_m))) * Float64(-0.125 / Float64(d * Float64(d * l))))), D_m, w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(D$95$m * N[(N[(h * N[(M$95$m * N[(w0 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied egg-rr 63.4%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \left(\left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right), D, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h, \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot M\_m}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (* w0 (fma h (* (* -0.125 (* D_m D_m)) (/ (* M_m M_m) (* d (* d l)))) 1.0))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = w0 * fma(h, ((-0.125 * (D_m * D_m)) * ((M_m * M_m) / (d * (d * l)))), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(w0 * fma(h, Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * M_m) / Float64(d * Float64(d * l)))), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(w0 * N[(h * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.0

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

   4. Applied egg-rr 61.3%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 59.8%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot \left(M\_m \cdot -0.125\right)\right) \cdot \left(h \cdot \frac{w0}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (* (* D_m D_m) (* (* M_m (* M_m -0.125)) (* h (/ w0 (* d (* d l))))))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (D_m * D_m) * ((M_m * (M_m * -0.125)) * (h * (w0 / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-5d+264)) then
        tmp = (d_m * d_m) * ((m_m * (m_m * (-0.125d0))) * (h * (w0 / (d * (d * l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (D_m * D_m) * ((M_m * (M_m * -0.125)) * (h * (w0 / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264:
		tmp = (D_m * D_m) * ((M_m * (M_m * -0.125)) * (h * (w0 / (d * (d * l)))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(M_m * Float64(M_m * -0.125)) * Float64(h * Float64(w0 / Float64(d * Float64(d * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+264)
		tmp = (D_m * D_m) * ((M_m * (M_m * -0.125)) * (h * (w0 / (d * (d * l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(M$95$m * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h * N[(w0 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 56.7%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 61.7%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative 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. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 53.8

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

   11. Simplified 53.8%

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

   13. Applied egg-rr 59.8%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\left(M \cdot \left(M \cdot -0.125\right)\right) \cdot \left(h \cdot \frac{w0}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \frac{\left(M\_m \cdot -0.125\right) \cdot \left(M\_m \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ h l)) -5e+264)
   (* (* D_m D_m) (/ (* (* M_m -0.125) (* M_m (* w0 h))) (* l (* d d))))
   w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (D_m * D_m) * (((M_m * -0.125) * (M_m * (w0 * h))) / (l * (d * d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-5d+264)) then
        tmp = (d_m * d_m) * (((m_m * (-0.125d0)) * (m_m * (w0 * h))) / (l * (d * d)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (D_m * D_m) * (((M_m * -0.125) * (M_m * (w0 * h))) / (l * (d * d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+264:
		tmp = (D_m * D_m) * (((M_m * -0.125) * (M_m * (w0 * h))) / (l * (d * d)))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m * -0.125) * Float64(M_m * Float64(w0 * h))) / Float64(l * Float64(d * d))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+264)
		tmp = (D_m * D_m) * (((M_m * -0.125) * (M_m * (w0 * h))) / (l * (d * d)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * -0.125), $MachinePrecision] * N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 56.7%

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

   6. Taylor expanded in w0 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 61.7%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative 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. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 53.8

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

   11. Simplified 53.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 56.7

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

   13. Applied egg-rr 56.7%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 91.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity 91.8

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{\left(M \cdot -0.125\right) \cdot \left(M \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{D\_m}{2 \cdot d} \cdot \left(M\_m \cdot \frac{\frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d} \cdot h}{\ell}\right)} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (* (/ D_m (* 2.0 d)) (* M_m (/ (* (/ (* D_m (* M_m 0.5)) d) h) l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((D_m / (2.0 * d)) * (M_m * ((((D_m * (M_m * 0.5)) / d) * h) / l)))));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * sqrt((1.0d0 - ((d_m / (2.0d0 * d)) * (m_m * ((((d_m * (m_m * 0.5d0)) / d) * h) / l)))))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((D_m / (2.0 * d)) * (M_m * ((((D_m * (M_m * 0.5)) / d) * h) / l)))));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * math.sqrt((1.0 - ((D_m / (2.0 * d)) * (M_m * ((((D_m * (M_m * 0.5)) / d) * h) / l)))))

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / Float64(2.0 * d)) * Float64(M_m * Float64(Float64(Float64(Float64(D_m * Float64(M_m * 0.5)) / d) * h) / l))))))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 - ((D_m / (2.0 * d)) * (M_m * ((((D_m * (M_m * 0.5)) / d) * h) / l)))));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*l/ N/A

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

   18. *-commutative N/A

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

4. Applied egg-rr 87.2%

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

Alternative 13: 67.7% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 68.9%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Step-by-step derivation

   1. *-rgt-identity 68.9

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

7. Applied egg-rr 68.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))))