Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.3%

Time: 10.7s

Alternatives: 14

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: 89.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.3%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 88.1%

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

Alternative 2: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+250}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
      1e+250)
   (* w0 1.0)
   (*
    w0
    (fma (* (* (* (/ h (* d d)) M_m) (/ M_m l)) D_m) (* -0.125 D_m) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+250) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(((((h / (d * d)) * M_m) * (M_m / l)) * D_m), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 1e+250)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * D_m), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+250], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $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))))) < 9.9999999999999992e249

   1. Initial program 91.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 77.4%

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

   if 9.9999999999999992e249 < (*.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 42.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 63.6%

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

Alternative 3: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\left(\frac{D\_m}{d} \cdot 0.25\right) \cdot M\_m\right) \cdot \left(M\_m \cdot \left(-h\right)\right), \frac{D\_m}{\ell \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M\_m}{d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e-7)
   (*
    w0
    (sqrt
     (fma (* (* (* (/ D_m d) 0.25) M_m) (* M_m (- h))) (/ D_m (* l d)) 1.0)))
   (*
    w0
    (fma (/ (* (/ (* h M_m) d) (* (/ M_m d) D_m)) l) (* -0.125 D_m) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-7) {
		tmp = w0 * sqrt(fma(((((D_m / d) * 0.25) * M_m) * (M_m * -h)), (D_m / (l * d)), 1.0));
	} else {
		tmp = w0 * fma(((((h * M_m) / d) * ((M_m / d) * D_m)) / l), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-7)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(D_m / d) * 0.25) * M_m) * Float64(M_m * Float64(-h))), Float64(D_m / Float64(l * d)), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M_m) / d) * Float64(Float64(M_m / d) * D_m)) / l), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-7], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * 0.25), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m * (-h)), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.9999999999999999e-7

   1. Initial program 65.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. 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 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 53.4

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

   6. Applied rewrites 53.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 54.9

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

   8. Applied rewrites 59.9%

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

   if -1.9999999999999999e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 67.2%

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

   9. Applied rewrites 91.8%

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

   11. Applied rewrites 92.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.25\right) \cdot M\right) \cdot \left(M \cdot \left(-h\right)\right), \frac{D}{\ell \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, -0.125 \cdot D, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-h\right) \cdot \frac{\left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot D\_m}{\ell \cdot d}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M\_m}{d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+115)
   (*
    w0
    (sqrt
     (fma (* (- h) (/ (* (* 0.25 (* M_m M_m)) D_m) (* l d))) (/ D_m d) 1.0)))
   (*
    w0
    (fma (/ (* (/ (* h M_m) d) (* (/ M_m d) D_m)) l) (* -0.125 D_m) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+115) {
		tmp = w0 * sqrt(fma((-h * (((0.25 * (M_m * M_m)) * D_m) / (l * d))), (D_m / d), 1.0));
	} else {
		tmp = w0 * fma(((((h * M_m) / d) * ((M_m / d) * D_m)) / l), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+115)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-h) * Float64(Float64(Float64(0.25 * Float64(M_m * M_m)) * D_m) / Float64(l * d))), Float64(D_m / d), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M_m) / d) * Float64(Float64(M_m / d) * D_m)) / l), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+115], N[(w0 * N[Sqrt[N[(N[((-h) * N[(N[(N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000008e115

   1. Initial program 62.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. 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 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 55.0

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

   6. Applied rewrites 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if -5.00000000000000008e115 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 66.1%

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

   9. Applied rewrites 90.0%

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

   11. Applied rewrites 91.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-h\right) \cdot \frac{\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot D}{\ell \cdot d}, \frac{D}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, -0.125 \cdot D, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.5%

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

   if 1 < (-.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 70.6%

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

   5. Taylor expanded in h around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. rgt-mult-inverse N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 56.0

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

   7. Applied rewrites 56.0%

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

Alternative 6: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+109}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M\_m}{d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+109)
   (*
    w0
    (sqrt (fma (* h -0.25) (/ (* (* (* D_m M_m) M_m) D_m) (* (* d d) l)) 1.0)))
   (*
    w0
    (fma (/ (* (/ (* h M_m) d) (* (/ M_m d) D_m)) l) (* -0.125 D_m) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+109) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((D_m * M_m) * M_m) * D_m) / ((d * d) * l)), 1.0));
	} else {
		tmp = w0 * fma(((((h * M_m) / d) * ((M_m / d) * D_m)) / l), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+109)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * D_m) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M_m) / d) * Float64(Float64(M_m / d) * D_m)) / l), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+109], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e109

   1. Initial program 62.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 62.8%

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

   5. Taylor expanded in h around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. rgt-mult-inverse N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   if -5.0000000000000001e109 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 66.4%

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

   9. Applied rewrites 90.5%

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

   11. Applied rewrites 91.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+109}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, -0.125 \cdot D, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.5%

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

   if 1 < (-.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. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 49.2%

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

Alternative 8: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \frac{\left(-0.125 \cdot \left(\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(\ell \cdot d\right) \cdot d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+245)
   (* w0 (/ (* (* -0.125 (* (* (* D_m M_m) M_m) D_m)) h) (* (* l d) d)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245) {
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((l * d) * d));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+245)) then
        tmp = w0 * ((((-0.125d0) * (((d_m * m_m) * m_m) * d_m)) * h) / ((l * d) * d))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245) {
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((l * d) * d));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245:
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((l * d) * d))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+245)
		tmp = Float64(w0 * Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D_m * M_m) * M_m) * D_m)) * h) / Float64(Float64(l * d) * d)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+245)
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((l * d) * d));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+245], N[(w0 * N[(N[(N[(-0.125 * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000009e245

   1. Initial program 57.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 50.4%

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

   10. Applied rewrites 50.7%

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

   if -2.00000000000000009e245 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

Alternative 9: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \frac{\left(-0.125 \cdot \left(\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+245)
   (* w0 (/ (* (* -0.125 (* (* (* D_m M_m) M_m) D_m)) h) (* (* d d) l)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245) {
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((d * d) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+245)) then
        tmp = w0 * ((((-0.125d0) * (((d_m * m_m) * m_m) * d_m)) * h) / ((d * d) * l))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245) {
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((d * d) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+245:
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((d * d) * l))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+245)
		tmp = Float64(w0 * Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D_m * M_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+245)
		tmp = w0 * (((-0.125 * (((D_m * M_m) * M_m) * D_m)) * h) / ((d * d) * l));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+245], N[(w0 * N[(N[(N[(-0.125 * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000009e245

   1. Initial program 57.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 50.4%

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

   if -2.00000000000000009e245 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

Alternative 10: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\frac{D\_m}{d} \cdot 0.25\right) \cdot M\_m\right) \cdot \left(M\_m \cdot h\right)}{-\ell}, \frac{D\_m}{d}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(D_m / d) * 0.25) * M_m) * Float64(M_m * h)) / Float64(-l)), Float64(D_m / d), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 80.3%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. 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 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 75.5

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 75.5

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

6. Applied rewrites 75.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 84.9

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

8. Applied rewrites 84.9%

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

9. Final simplification 84.9%

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

Alternative 11: 83.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M\_m}{d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell}, -0.125 \cdot D\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot -0.25}{d \cdot \ell}, \frac{D\_m}{d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 8.8e-141)
   (* w0 (fma (/ (* (/ (* h M_m) d) (* (/ M_m d) D_m)) l) (* -0.125 D_m) 1.0))
   (*
    w0
    (sqrt
     (fma (/ (* (* (* (* M_m M_m) h) D_m) -0.25) (* d l)) (/ D_m d) 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 8.8e-141) {
		tmp = w0 * fma(((((h * M_m) / d) * ((M_m / d) * D_m)) / l), (-0.125 * D_m), 1.0);
	} else {
		tmp = w0 * sqrt(fma((((((M_m * M_m) * h) * D_m) * -0.25) / (d * l)), (D_m / d), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 8.8e-141)
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M_m) / d) * Float64(Float64(M_m / d) * D_m)) / l), Float64(-0.125 * D_m), 1.0));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * -0.25) / Float64(d * l)), Float64(D_m / d), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 8.8e-141], N[(w0 * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * -0.25), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.80000000000000037e-141

   1. Initial program 83.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 84.8%

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

   11. Applied rewrites 85.9%

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

   if 8.80000000000000037e-141 < M

   1. Initial program 74.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 64.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.1

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.9

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

   9. Applied rewrites 65.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h \cdot M}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{\ell}, -0.125 \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot -0.25}{d \cdot \ell}, \frac{D}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-263}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M\_m}{\ell \cdot d} \cdot \frac{M\_m}{d}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e-263)
   (* w0 1.0)
   (*
    w0
    (fma (* (* (/ (* h M_m) (* l d)) (/ M_m d)) D_m) (* -0.125 D_m) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-263) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(((((h * M_m) / (l * d)) * (M_m / d)) * D_m), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-263)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M_m) / Float64(l * d)) * Float64(M_m / d)) * D_m), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-263], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2e-263

   1. Initial program 81.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 73.1%

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

   if 2e-263 < (*.f64 M D)

   1. Initial program 78.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 76.5%

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

   11. Applied rewrites 74.4%

       \[\leadsto w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{\ell \cdot d} \cdot \frac{M}{d}\right) \cdot D, -0.125 \cdot D, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, D\_m \cdot \frac{-0.125 \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell}}{d \cdot d}, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 5.2e-109)
   (* w0 1.0)
   (*
    w0
    (fma D_m (* D_m (/ (* -0.125 (/ (* (* M_m M_m) h) l)) (* d d))) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5.2e-109) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(D_m, (D_m * ((-0.125 * (((M_m * M_m) * h) / l)) / (d * d))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 5.2e-109)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(D_m, Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * M_m) * h) / l)) / Float64(d * d))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 5.2e-109], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.1999999999999997e-109

   1. Initial program 83.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 78.4%

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

   if 5.1999999999999997e-109 < M

   1. Initial program 73.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 53.9%

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

Alternative 14: 68.2% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.8%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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))))))