Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.9%

Time: 13.5s

Alternatives: 15

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: 81.3% 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: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\\ \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d\_m} \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d\_m}}{\ell} \cdot t\_0, h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0, \frac{D\_m \cdot 0.5}{-\ell} \cdot \left(h \cdot \frac{M\_m}{d\_m}\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* D_m (* M_m (/ 0.5 d_m)))))
   (if (<= (/ (* D_m M_m) (* 2.0 d_m)) 4e+70)
     (* (sqrt (fma (* (/ (/ (* -0.5 (* D_m M_m)) d_m) l) t_0) h 1.0)) w0)
     (* (sqrt (fma t_0 (* (/ (* D_m 0.5) (- l)) (* h (/ M_m d_m))) 1.0)) w0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m * (M_m * (0.5 / d_m));
	double tmp;
	if (((D_m * M_m) / (2.0 * d_m)) <= 4e+70) {
		tmp = sqrt(fma(((((-0.5 * (D_m * M_m)) / d_m) / l) * t_0), h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma(t_0, (((D_m * 0.5) / -l) * (h * (M_m / d_m))), 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d_m)))
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) <= 4e+70)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d_m) / l) * t_0), h, 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(t_0, Float64(Float64(Float64(D_m * 0.5) / Float64(-l)) * Float64(h * Float64(M_m / d_m))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 4e+70], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$0), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] / (-l)), $MachinePrecision] * N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < 4.00000000000000029e70

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      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 88.1%

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

   6. Applied rewrites 89.9%

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

   if 4.00000000000000029e70 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      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 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 52.4

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

   6. Applied rewrites 52.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{2 \cdot d} \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{-0.5 \cdot \left(D \cdot M\right)}{d}}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right), h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{0.5}{d}\right), \frac{D \cdot 0.5}{-\ell} \cdot \left(h \cdot \frac{M}{d}\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2}} \cdot w0 \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\_m\right) \cdot \left(\frac{D\_m}{d\_m} \cdot M\_m\right)\right) \cdot \frac{-h}{\ell}, \frac{D\_m}{d\_m}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m, \left(\left(\frac{-0.5 \cdot D\_m}{\ell} \cdot \frac{M\_m}{d\_m}\right) \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\right) \cdot h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (* (sqrt (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)))) w0)
      5e+222)
   (*
    (sqrt
     (fma
      (* (* (* 0.25 M_m) (* (/ D_m d_m) M_m)) (/ (- h) l))
      (/ D_m d_m)
      1.0))
    w0)
   (*
    (sqrt
     (fma
      D_m
      (* (* (* (/ (* -0.5 D_m) l) (/ M_m d_m)) (* M_m (/ 0.5 d_m))) h)
      1.0))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((sqrt((1.0 - ((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)))) * w0) <= 5e+222) {
		tmp = sqrt(fma((((0.25 * M_m) * ((D_m / d_m) * M_m)) * (-h / l)), (D_m / d_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(D_m, (((((-0.5 * D_m) / l) * (M_m / d_m)) * (M_m * (0.5 / d_m))) * h), 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)))) * w0) <= 5e+222)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.25 * M_m) * Float64(Float64(D_m / d_m) * M_m)) * Float64(Float64(-h) / l)), Float64(D_m / d_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(D_m, Float64(Float64(Float64(Float64(Float64(-0.5 * D_m) / l) * Float64(M_m / d_m)) * Float64(M_m * Float64(0.5 / d_m))) * h), 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], 5e+222], N[(N[Sqrt[N[(N[(N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(D$95$m * N[(N[(N[(N[(N[(-0.5 * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 5.00000000000000023e222

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.1

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

   6. Applied rewrites 85.1%

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

   if 5.00000000000000023e222 < (*.f64 w0 (sqrt.f64 (-.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 48.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 70.2%

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

   6. Applied rewrites 67.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 72.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \frac{-h}{\ell}, \frac{D}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D, \left(\left(\frac{-0.5 \cdot D}{\ell} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot h, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2}} \cdot w0 \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\_m\right) \cdot \left(\frac{D\_m}{d\_m} \cdot M\_m\right)\right) \cdot \frac{-h}{\ell}, \frac{D\_m}{d\_m}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \left(-0.5 \cdot D\_m\right)\right) \cdot \left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\right), h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (* (sqrt (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)))) w0)
      2e+202)
   (*
    (sqrt
     (fma
      (* (* (* 0.25 M_m) (* (/ D_m d_m) M_m)) (/ (- h) l))
      (/ D_m d_m)
      1.0))
    w0)
   (*
    (sqrt
     (fma
      (* (* (/ M_m (* l d_m)) (* -0.5 D_m)) (* D_m (* M_m (/ 0.5 d_m))))
      h
      1.0))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((sqrt((1.0 - ((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)))) * w0) <= 2e+202) {
		tmp = sqrt(fma((((0.25 * M_m) * ((D_m / d_m) * M_m)) * (-h / l)), (D_m / d_m), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((M_m / (l * d_m)) * (-0.5 * D_m)) * (D_m * (M_m * (0.5 / d_m)))), h, 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)))) * w0) <= 2e+202)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.25 * M_m) * Float64(Float64(D_m / d_m) * M_m)) * Float64(Float64(-h) / l)), Float64(D_m / d_m), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(-0.5 * D_m)) * Float64(D_m * Float64(M_m * Float64(0.5 / d_m)))), h, 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], 2e+202], N[(N[Sqrt[N[(N[(N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 1.9999999999999998e202

   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. 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. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   if 1.9999999999999998e202 < (*.f64 w0 (sqrt.f64 (-.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 49.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 71.0%

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

   6. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 69.6

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

   8. Applied rewrites 69.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}} \cdot w0 \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(0.25 \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \frac{-h}{\ell}, \frac{D}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{M}{\ell \cdot d} \cdot \left(-0.5 \cdot D\right)\right) \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right), h, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{\frac{\frac{D\_m}{d\_m}}{d\_m}}{\ell} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot M\_m\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -1e+27)
   (*
    (sqrt (* (* -0.25 h) (* (* (/ (/ (/ D_m d_m) d_m) l) (* D_m M_m)) M_m)))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = sqrt(((-0.25 * h) * (((((D_m / d_m) / d_m) / l) * (D_m * M_m)) * M_m))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-1d+27)) then
        tmp = sqrt((((-0.25d0) * h) * (((((d_m / d_m_1) / d_m_1) / l) * (d_m * m_m)) * m_m))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = Math.sqrt(((-0.25 * h) * (((((D_m / d_m) / d_m) / l) * (D_m * M_m)) * M_m))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27:
		tmp = math.sqrt(((-0.25 * h) * (((((D_m / d_m) / d_m) / l) * (D_m * M_m)) * M_m))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = Float64(sqrt(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(Float64(Float64(D_m / d_m) / d_m) / l) * Float64(D_m * M_m)) * M_m))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = sqrt(((-0.25 * h) * (((((D_m / d_m) / d_m) / l) * (D_m * M_m)) * M_m))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+27], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $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)) < -1e27

   1. Initial program 60.5%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 49.0%

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

   if -1e27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{\frac{\frac{D}{d}}{d}}{\ell} \cdot \left(D \cdot M\right)\right) \cdot M\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{\left(D\_m \cdot M\_m\right) \cdot D\_m}{\ell \cdot d\_m}}{d\_m} \cdot M\_m\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -1e+27)
   (*
    (sqrt (* (* (/ (/ (* (* D_m M_m) D_m) (* l d_m)) d_m) M_m) (* -0.25 h)))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = sqrt(((((((D_m * M_m) * D_m) / (l * d_m)) / d_m) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-1d+27)) then
        tmp = sqrt(((((((d_m * m_m) * d_m) / (l * d_m_1)) / d_m_1) * m_m) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = Math.sqrt(((((((D_m * M_m) * D_m) / (l * d_m)) / d_m) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27:
		tmp = math.sqrt(((((((D_m * M_m) * D_m) / (l * d_m)) / d_m) * M_m) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * D_m) / Float64(l * d_m)) / d_m) * M_m) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = sqrt(((((((D_m * M_m) * D_m) / (l * d_m)) / d_m) * M_m) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+27], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.25 * h), $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)) < -1e27

   1. Initial program 60.5%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 40.4%

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 52.2%

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

   if -1e27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{\left(D \cdot M\right) \cdot D}{\ell \cdot d}}{d} \cdot M\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D\_m}{\left(\ell \cdot d\_m\right) \cdot d\_m} \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -1e+27)
   (*
    (sqrt (* (* (* (/ D_m (* (* l d_m) d_m)) M_m) (* D_m M_m)) (* -0.25 h)))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = sqrt(((((D_m / ((l * d_m) * d_m)) * M_m) * (D_m * M_m)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-1d+27)) then
        tmp = sqrt(((((d_m / ((l * d_m_1) * d_m_1)) * m_m) * (d_m * m_m)) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = Math.sqrt(((((D_m / ((l * d_m) * d_m)) * M_m) * (D_m * M_m)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27:
		tmp = math.sqrt(((((D_m / ((l * d_m) * d_m)) * M_m) * (D_m * M_m)) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D_m / Float64(Float64(l * d_m) * d_m)) * M_m) * Float64(D_m * M_m)) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = sqrt(((((D_m / ((l * d_m) * d_m)) * M_m) * (D_m * M_m)) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+27], N[(N[Sqrt[N[(N[(N[(N[(D$95$m / N[(N[(l * d$95$m), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $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)) < -1e27

   1. Initial program 60.5%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 40.4%

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

   9. Applied rewrites 49.3%

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

   if -1e27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D}{\left(\ell \cdot d\right) \cdot d} \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D\_m}{\left(\ell \cdot d\_m\right) \cdot d\_m} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot M\_m\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -1e+27)
   (*
    (sqrt (* (* (* (/ D_m (* (* l d_m) d_m)) (* D_m M_m)) M_m) (* -0.25 h)))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = sqrt(((((D_m / ((l * d_m) * d_m)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-1d+27)) then
        tmp = sqrt(((((d_m / ((l * d_m_1) * d_m_1)) * (d_m * m_m)) * m_m) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27) {
		tmp = Math.sqrt(((((D_m / ((l * d_m) * d_m)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -1e+27:
		tmp = math.sqrt(((((D_m / ((l * d_m) * d_m)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D_m / Float64(Float64(l * d_m) * d_m)) * Float64(D_m * M_m)) * M_m) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d_m)) ^ 2.0)) <= -1e+27)
		tmp = sqrt(((((D_m / ((l * d_m) * d_m)) * (D_m * M_m)) * M_m) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+27], N[(N[Sqrt[N[(N[(N[(N[(D$95$m / N[(N[(l * d$95$m), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.25 * h), $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)) < -1e27

   1. Initial program 60.5%

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

   3. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 40.4%

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

   9. Applied rewrites 49.2%

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

   if -1e27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 95.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D}{\left(\ell \cdot d\right) \cdot d} \cdot \left(D \cdot M\right)\right) \cdot M\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(D\_m \cdot D\_m\right) \cdot h\right) \cdot M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{d\_m}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -5e+67)
   (fma
    (* -0.125 w0)
    (* (/ (* (* (* D_m D_m) h) M_m) (* l d_m)) (/ M_m d_m))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -5e+67) {
		tmp = fma((-0.125 * w0), (((((D_m * D_m) * h) * M_m) / (l * d_m)) * (M_m / d_m)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -5e+67)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(Float64(D_m * D_m) * h) * M_m) / Float64(l * d_m)) * Float64(M_m / d_m)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+67], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d$95$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)) < -4.99999999999999976e67

   1. Initial program 58.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 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 28.8%

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

   9. Taylor expanded in w0 around 0

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

   11. Applied rewrites 30.7%

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

   13. Applied rewrites 41.8%

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

   if -4.99999999999999976e67 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 92.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot M}{\ell \cdot d} \cdot \frac{M}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{M\_m}{\ell} \cdot \frac{\left(\left(D\_m \cdot D\_m\right) \cdot h\right) \cdot M\_m}{d\_m \cdot d\_m}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -5e+58)
   (fma
    (* -0.125 w0)
    (* (/ M_m l) (/ (* (* (* D_m D_m) h) M_m) (* d_m d_m)))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -5e+58) {
		tmp = fma((-0.125 * w0), ((M_m / l) * ((((D_m * D_m) * h) * M_m) / (d_m * d_m))), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -5e+58)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(M_m / l) * Float64(Float64(Float64(Float64(D_m * D_m) * h) * M_m) / Float64(d_m * d_m))), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+58], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(d$95$m * d$95$m), $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)) < -4.99999999999999986e58

   1. Initial program 59.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.9%

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

   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 28.3%

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

   9. Taylor expanded in w0 around 0

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

   11. Applied rewrites 30.0%

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

   13. Applied rewrites 35.2%

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

   if -4.99999999999999986e58 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 93.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{M}{\ell} \cdot \frac{\left(\left(D \cdot D\right) \cdot h\right) \cdot M}{d \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d\_m}\right)}^{2} \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{M\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot h\right) \cdot M\_m\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d_m)) 2.0)) -5e+67)
   (fma
    (* -0.125 w0)
    (* (/ M_m (* (* d_m d_m) l)) (* (* (* D_m D_m) h) M_m))
    w0)
   (* 1.0 w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d_m)), 2.0)) <= -5e+67) {
		tmp = fma((-0.125 * w0), ((M_m / ((d_m * d_m) * l)) * (((D_m * D_m) * h) * M_m)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -5e+67)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(M_m / Float64(Float64(d_m * d_m) * l)) * Float64(Float64(Float64(D_m * D_m) * h) * M_m)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+67], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(M$95$m / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $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)) < -4.99999999999999976e67

   1. Initial program 58.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 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 28.8%

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

   9. Taylor expanded in w0 around 0

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

   11. Applied rewrites 30.7%

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

   13. Applied rewrites 35.7%

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

   if -4.99999999999999976e67 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 92.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{M}{\left(d \cdot d\right) \cdot \ell} \cdot \left(\left(\left(D \cdot D\right) \cdot h\right) \cdot M\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d\_m}\\ \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d\_m}}{\ell} \cdot \left(D\_m \cdot t\_0\right), h, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m, \left(\left(\frac{-0.5 \cdot D\_m}{\ell} \cdot \frac{M\_m}{d\_m}\right) \cdot t\_0\right) \cdot h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d_m))))
   (if (<= (/ (* D_m M_m) (* 2.0 d_m)) 2e+17)
     (*
      (sqrt (fma (* (/ (/ (* -0.5 (* D_m M_m)) d_m) l) (* D_m t_0)) h 1.0))
      w0)
     (*
      (sqrt (fma D_m (* (* (* (/ (* -0.5 D_m) l) (/ M_m d_m)) t_0) h) 1.0))
      w0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (0.5 / d_m);
	double tmp;
	if (((D_m * M_m) / (2.0 * d_m)) <= 2e+17) {
		tmp = sqrt(fma(((((-0.5 * (D_m * M_m)) / d_m) / l) * (D_m * t_0)), h, 1.0)) * w0;
	} else {
		tmp = sqrt(fma(D_m, (((((-0.5 * D_m) / l) * (M_m / d_m)) * t_0) * h), 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(M_m * Float64(0.5 / d_m))
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(2.0 * d_m)) <= 2e+17)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d_m) / l) * Float64(D_m * t_0)), h, 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(D_m, Float64(Float64(Float64(Float64(Float64(-0.5 * D_m) / l) * Float64(M_m / d_m)) * t_0) * h), 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2e+17], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(D$95$m * N[(N[(N[(N[(N[(-0.5 * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * h), $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)) < 2e17

   1. Initial program 83.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 88.6%

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

   6. Applied rewrites 90.5%

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

   if 2e17 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 55.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 56.0%

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

   6. Applied rewrites 52.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 54.1%

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

4. Final simplification 82.8%

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

Alternative 12: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d\_m \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right), \frac{\left(-0.5 \cdot \left(D\_m \cdot M\_m\right)\right) \cdot h}{\ell \cdot d\_m}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell} \cdot \left(-0.25 \cdot \frac{D\_m}{d\_m}\right), \frac{D\_m}{d\_m}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* 2.0 d_m) 5e+59)
   (*
    (sqrt
     (fma
      (* D_m (* M_m (/ 0.5 d_m)))
      (/ (* (* -0.5 (* D_m M_m)) h) (* l d_m))
      1.0))
    w0)
   (*
    (sqrt
     (fma (* (/ (* (* M_m M_m) h) l) (* -0.25 (/ D_m d_m))) (/ D_m d_m) 1.0))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((2.0 * d_m) <= 5e+59) {
		tmp = sqrt(fma((D_m * (M_m * (0.5 / d_m))), (((-0.5 * (D_m * M_m)) * h) / (l * d_m)), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((((M_m * M_m) * h) / l) * (-0.25 * (D_m / d_m))), (D_m / d_m), 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(2.0 * d_m) <= 5e+59)
		tmp = Float64(sqrt(fma(Float64(D_m * Float64(M_m * Float64(0.5 / d_m))), Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) * h) / Float64(l * d_m)), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m * M_m) * h) / l) * Float64(-0.25 * Float64(D_m / d_m))), Float64(D_m / d_m), 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(2.0 * d$95$m), $MachinePrecision], 5e+59], N[(N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 4.9999999999999997e59

   1. Initial program 78.3%

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

      1. lift--.f64 N/A

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

      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 78.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

   6. Applied rewrites 74.8%

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

   if 4.9999999999999997e59 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 75.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in h around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{0.5}{d}\right), \frac{\left(-0.5 \cdot \left(D \cdot M\right)\right) \cdot h}{\ell \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot \left(-0.25 \cdot \frac{D}{d}\right), \frac{D}{d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. lift--.f64 N/A

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

   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 81.7%

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

5. Final simplification 81.7%

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

Alternative 14: 85.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. lift--.f64 N/A

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

   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 81.7%

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

6. Applied rewrites 82.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 78.9

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

8. Applied rewrites 78.9%

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

9. Final simplification 78.9%

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

Alternative 15: 68.6% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = 1.0d0 * w0
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return 1.0 * w0;
}

Python

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

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(1.0 * w0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 62.3%

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

6. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2024273 
(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))))))