Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 89.0%

Time: 13.9s

Alternatives: 12

Speedup: 2.1×

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: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq 10^{+267}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d} \cdot \frac{h}{\ell}, D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{M\_m \cdot 0.5}{\ell \cdot d} \cdot D\_m, \left(\left(-D\_m\right) \cdot 0.5\right) \cdot \left(h \cdot \frac{M\_m}{d}\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0))) 1e+267)
   (*
    (sqrt
     (fma
      (* (/ (* -0.5 (* D_m M_m)) d) (/ h l))
      (* D_m (* M_m (/ 0.5 d)))
      1.0))
    w0)
   (*
    (sqrt
     (fma
      (* (/ (* M_m 0.5) (* l d)) D_m)
      (* (* (- D_m) 0.5) (* h (/ M_m d)))
      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 ((1.0 - ((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0))) <= 1e+267) {
		tmp = sqrt(fma((((-0.5 * (D_m * M_m)) / d) * (h / l)), (D_m * (M_m * (0.5 / d))), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((M_m * 0.5) / (l * d)) * D_m), ((-D_m * 0.5) * (h * (M_m / d))), 1.0)) * 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(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0))) <= 1e+267)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d) * Float64(h / l)), Float64(D_m * Float64(M_m * Float64(0.5 / d))), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m * 0.5) / Float64(l * d)) * D_m), Float64(Float64(Float64(-D_m) * 0.5) * Float64(h * Float64(M_m / d))), 1.0)) * 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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+267], N[(N[Sqrt[N[(N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[((-D$95$m) * 0.5), $MachinePrecision] * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 9.9999999999999997e266

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 98.3%

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

   if 9.9999999999999997e266 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.3

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 71.7

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

   10. Applied rewrites 71.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq 10^{+267}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(D \cdot M\right)}{d} \cdot \frac{h}{\ell}, D \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{M \cdot 0.5}{\ell \cdot d} \cdot D, \left(\left(-D\right) \cdot 0.5\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.7× speedup

Math

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

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -0.5)
   (*
    (sqrt (- 1.0 (* (* (/ M_m (* l d)) (* h (/ M_m d))) (* (* D_m D_m) 0.25))))
    w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5) {
		tmp = sqrt((1.0 - (((M_m / (l * d)) * (h * (M_m / d))) * ((D_m * D_m) * 0.25)))) * w0;
	} else {
		tmp = 1.0 * 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 (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-0.5d0)) then
        tmp = sqrt((1.0d0 - (((m_m / (l * d)) * (h * (m_m / d))) * ((d_m * d_m) * 0.25d0)))) * w0
    else
        tmp = 1.0d0 * 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 (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5) {
		tmp = Math.sqrt((1.0 - (((M_m / (l * d)) * (h * (M_m / d))) * ((D_m * D_m) * 0.25)))) * w0;
	} else {
		tmp = 1.0 * 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 ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5:
		tmp = math.sqrt((1.0 - (((M_m / (l * d)) * (h * (M_m / d))) * ((D_m * D_m) * 0.25)))) * w0
	else:
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -0.5)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(M_m / Float64(l * d)) * Float64(h * Float64(M_m / d))) * Float64(Float64(D_m * D_m) * 0.25)))) * w0);
	else
		tmp = Float64(1.0 * 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 (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -0.5)
		tmp = sqrt((1.0 - (((M_m / (l * d)) * (h * (M_m / d))) * ((D_m * D_m) * 0.25)))) * w0;
	else
		tmp = 1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.5

   1. Initial program 72.3%

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

   3. Taylor expanded in h around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 54.3%

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

   if -0.5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

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

Alternative 3: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\frac{h \cdot M\_m}{\ell \cdot d} \cdot \frac{M\_m}{d}\right) \cdot \left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -1e+52)
   (* (sqrt (* (* (/ (* h M_m) (* l d)) (/ M_m d)) (* -0.25 (* D_m D_m)))) w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+52) {
		tmp = sqrt(((((h * M_m) / (l * d)) * (M_m / d)) * (-0.25 * (D_m * D_m)))) * w0;
	} else {
		tmp = 1.0 * 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 (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-1d+52)) then
        tmp = sqrt(((((h * m_m) / (l * d)) * (m_m / d)) * ((-0.25d0) * (d_m * d_m)))) * w0
    else
        tmp = 1.0d0 * 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 (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+52) {
		tmp = Math.sqrt(((((h * M_m) / (l * d)) * (M_m / d)) * (-0.25 * (D_m * D_m)))) * w0;
	} else {
		tmp = 1.0 * 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 ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+52:
		tmp = math.sqrt(((((h * M_m) / (l * d)) * (M_m / d)) * (-0.25 * (D_m * D_m)))) * w0
	else:
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+52)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(h * M_m) / Float64(l * d)) * Float64(M_m / d)) * Float64(-0.25 * Float64(D_m * D_m)))) * w0);
	else
		tmp = Float64(1.0 * 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 (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -1e+52)
		tmp = sqrt(((((h * M_m) / (l * d)) * (M_m / d)) * (-0.25 * (D_m * D_m)))) * w0;
	else
		tmp = 1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+52], N[(N[Sqrt[N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999999e51

   1. Initial program 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in h around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 43.8

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

   7. Applied rewrites 43.8%

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

   9. Applied rewrites 56.5%

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

   if -9.9999999999999999e51 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.9%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\frac{h \cdot M}{\ell \cdot d} \cdot \frac{M}{d}\right) \cdot \left(-0.25 \cdot \left(D \cdot D\right)\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.8× speedup

Math

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

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -0.5)
   (*
    (sqrt (- 1.0 (* (* (* (/ h (* (* d d) l)) 0.25) (* D_m M_m)) (* D_m M_m))))
    w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5) {
		tmp = sqrt((1.0 - ((((h / ((d * d) * l)) * 0.25) * (D_m * M_m)) * (D_m * M_m)))) * w0;
	} else {
		tmp = 1.0 * 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 (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-0.5d0)) then
        tmp = sqrt((1.0d0 - ((((h / ((d * d) * l)) * 0.25d0) * (d_m * m_m)) * (d_m * m_m)))) * w0
    else
        tmp = 1.0d0 * 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 (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5) {
		tmp = Math.sqrt((1.0 - ((((h / ((d * d) * l)) * 0.25) * (D_m * M_m)) * (D_m * M_m)))) * w0;
	} else {
		tmp = 1.0 * 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 ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -0.5:
		tmp = math.sqrt((1.0 - ((((h / ((d * d) * l)) * 0.25) * (D_m * M_m)) * (D_m * M_m)))) * w0
	else:
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -0.5)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h / Float64(Float64(d * d) * l)) * 0.25) * Float64(D_m * M_m)) * Float64(D_m * M_m)))) * w0);
	else
		tmp = Float64(1.0 * 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 (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -0.5)
		tmp = sqrt((1.0 - ((((h / ((d * d) * l)) * 0.25) * (D_m * M_m)) * (D_m * M_m)))) * w0;
	else
		tmp = 1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.5

   1. Initial program 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. clear-num N/A

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

      23. lower-/.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if -0.5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

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

Alternative 5: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{h \cdot w0}{\ell \cdot d} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+184)
   (fma (* -0.125 (* D_m D_m)) (* (/ (* h w0) (* l d)) (* (/ M_m d) M_m)) w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+184) {
		tmp = fma((-0.125 * (D_m * D_m)), (((h * w0) / (l * d)) * ((M_m / d) * M_m)), w0);
	} else {
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+184)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(h * w0) / Float64(l * d)) * Float64(Float64(M_m / d) * M_m)), w0);
	else
		tmp = Float64(1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+184], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * w0), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000003e184

   1. Initial program 67.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   7. 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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 43.2%

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

   10. Applied rewrites 47.5%

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

   12. Applied rewrites 53.7%

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

   if -2.00000000000000003e184 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot w0}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot M\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot M\_m\right) \cdot \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right), \frac{w0}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+217)
   (fma (* (* (* (/ h (* d d)) M_m) M_m) (* -0.125 (* D_m D_m))) (/ w0 l) w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+217) {
		tmp = fma(((((h / (d * d)) * M_m) * M_m) * (-0.125 * (D_m * D_m))), (w0 / l), w0);
	} else {
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+217)
		tmp = fma(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * M_m) * Float64(-0.125 * Float64(D_m * D_m))), Float64(w0 / l), w0);
	else
		tmp = Float64(1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+217], N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / l), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999992e217

   1. Initial program 65.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   7. 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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 45.7%

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

   10. Applied rewrites 50.2%

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

   12. Applied rewrites 50.7%

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

   if -1.99999999999999992e217 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 88.3%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{h}{d \cdot d} \cdot M\right) \cdot M\right) \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right), \frac{w0}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(\left(h \cdot w0\right) \cdot M\_m\right) \cdot M\_m}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+217)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* (* h w0) M_m) M_m) (* (* d d) l)) w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+217) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((h * w0) * M_m) * M_m) / ((d * d) * l)), w0);
	} else {
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+217)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(h * w0) * M_m) * M_m) / Float64(Float64(d * d) * l)), w0);
	else
		tmp = Float64(1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+217], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * w0), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999992e217

   1. Initial program 65.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   7. 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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 45.7%

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

   10. Applied rewrites 50.2%

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

   12. Applied rewrites 50.3%

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

   if -1.99999999999999992e217 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 88.3%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(\left(h \cdot w0\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{h \cdot w0}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M\_m \cdot M\_m\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -5e+266)
   (fma (* -0.125 (* D_m D_m)) (* (/ (* h w0) (* (* d d) l)) (* M_m M_m)) w0)
   (* 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 (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -5e+266) {
		tmp = fma((-0.125 * (D_m * D_m)), (((h * w0) / ((d * d) * l)) * (M_m * M_m)), w0);
	} else {
		tmp = 1.0 * 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(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -5e+266)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(h * w0) / Float64(Float64(d * d) * l)) * Float64(M_m * M_m)), w0);
	else
		tmp = Float64(1.0 * 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[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+266], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * w0), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.9999999999999999e266

   1. Initial program 63.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   6. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   7. 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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 54.2%

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

   12. Applied rewrites 55.7%

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

   if -4.9999999999999999e266 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 86.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{h \cdot w0}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d} \leq 2 \cdot 10^{-43}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left(D\_m \cdot 0.5\right), \frac{-0.5 \cdot h}{\ell \cdot d} \cdot D\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 (<= (/ (* D_m M_m) (* 2.0 d)) 2e-43)
   (* 1.0 w0)
   (*
    (sqrt
     (fma
      (* (* (/ M_m d) M_m) (* D_m 0.5))
      (* (/ (* -0.5 h) (* l d)) D_m)
      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 (((D_m * M_m) / (2.0 * d)) <= 2e-43) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma((((M_m / d) * M_m) * (D_m * 0.5)), (((-0.5 * h) / (l * d)) * D_m), 1.0)) * 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(D_m * M_m) / Float64(2.0 * d)) <= 2e-43)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m / d) * M_m) * Float64(D_m * 0.5)), Float64(Float64(Float64(-0.5 * h) / Float64(l * d)) * D_m), 1.0)) * 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[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e-43], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.00000000000000015e-43

   1. Initial program 86.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 79.3%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 2.00000000000000015e-43 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 71.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. associate-*l* N/A

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 64.6

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

   6. Applied rewrites 64.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 67.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{2 \cdot d} \leq 2 \cdot 10^{-43}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{M}{d} \cdot M\right) \cdot \left(D \cdot 0.5\right), \frac{-0.5 \cdot h}{\ell \cdot d} \cdot D, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.1% accurate, 1.9× speedup

Math

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

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
 (*
  (sqrt
   (fma
    (* D_m (* M_m (/ 0.5 d)))
    (/ (* (* h (/ M_m d)) (* D_m 0.5)) (- 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) {
	return sqrt(fma((D_m * (M_m * (0.5 / d))), (((h * (M_m / d)) * (D_m * 0.5)) / -l), 1.0)) * 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 Float64(sqrt(fma(Float64(D_m * Float64(M_m * Float64(0.5 / d))), Float64(Float64(Float64(h * Float64(M_m / d)) * Float64(D_m * 0.5)) / Float64(-l)), 1.0)) * 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_] := N[(N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

5. Final simplification 87.6%

   \[\leadsto \sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{0.5}{d}\right), \frac{\left(h \cdot \frac{M}{d}\right) \cdot \left(D \cdot 0.5\right)}{-\ell}, 1\right)} \cdot w0 \]
6. Add Preprocessing

Alternative 11: 85.5% accurate, 2.1× speedup

Math

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

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
 (*
  (sqrt
   (fma
    (* (/ (* M_m 0.5) (* l d)) D_m)
    (* (* (- D_m) 0.5) (* h (/ M_m d)))
    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) {
	return sqrt(fma((((M_m * 0.5) / (l * d)) * D_m), ((-D_m * 0.5) * (h * (M_m / d))), 1.0)) * 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 Float64(sqrt(fma(Float64(Float64(Float64(M_m * 0.5) / Float64(l * d)) * D_m), Float64(Float64(Float64(-D_m) * 0.5) * Float64(h * Float64(M_m / d))), 1.0)) * 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_] := N[(N[Sqrt[N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[((-D$95$m) * 0.5), $MachinePrecision] * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

6. Applied rewrites 87.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 87.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 87.6

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 87.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 87.6

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

8. Applied rewrites 87.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 84.2

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

10. Applied rewrites 84.2%

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

11. Final simplification 84.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(\frac{M \cdot 0.5}{\ell \cdot d} \cdot D, \left(\left(-D\right) \cdot 0.5\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right)} \cdot w0 \]
12. Add Preprocessing

Alternative 12: 67.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ 1 \cdot w0 \end{array} \]

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 (* 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) {
	return 1.0 * 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 = 1.0d0 * 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 1.0 * 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 1.0 * 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 Float64(1.0 * 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 = 1.0 * 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_] := N[(1.0 * w0), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 66.5%

   \[\leadsto w0 \cdot \color{blue}{1} \]

6. Final simplification 66.5%

   \[\leadsto 1 \cdot w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024251 
(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))))))