Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 89.3%

Time: 11.6s

Alternatives: 17

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 80.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

Alternative 2: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot \frac{D\_m}{d}\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d}, \left(\frac{-h}{\ell} \cdot M\_m\right) \cdot \left(\left(0.25 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, \frac{\left(\left(D\_m \cdot 0.5\right) \cdot h\right) \cdot M\_m}{\left(-d\right) \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)))
   (if (<= t_0 0.0)
     (*
      w0
      (sqrt
       (fma (* (* -0.25 (/ D_m d)) (/ (* (* M_m M_m) h) l)) (/ D_m d) 1.0)))
     (if (<= t_0 4e+125)
       (*
        w0
        (sqrt
         (fma
          (/ D_m d)
          (* (* (/ (- h) l) M_m) (* (* 0.25 M_m) (/ D_m d)))
          1.0)))
       (*
        w0
        (sqrt
         (fma
          (* (* (/ 0.5 d) M_m) D_m)
          (/ (* (* (* D_m 0.5) h) M_m) (* (- 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 t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = w0 * sqrt(fma(((-0.25 * (D_m / d)) * (((M_m * M_m) * h) / l)), (D_m / d), 1.0));
	} else if (t_0 <= 4e+125) {
		tmp = w0 * sqrt(fma((D_m / d), (((-h / l) * M_m) * ((0.25 * M_m) * (D_m / d))), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((0.5 / d) * M_m) * D_m), ((((D_m * 0.5) * h) * M_m) / (-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)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(D_m / d)) * Float64(Float64(Float64(M_m * M_m) * h) / l)), Float64(D_m / d), 1.0)));
	elseif (t_0 <= 4e+125)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d), Float64(Float64(Float64(Float64(-h) / l) * M_m) * Float64(Float64(0.25 * M_m) * Float64(D_m / d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d) * M_m) * D_m), Float64(Float64(Float64(Float64(D_m * 0.5) * h) * M_m) / Float64(Float64(-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_] := Block[{t$95$0 = N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+125], N[(w0 * N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[((-h) / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] / N[((-d) * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 0.0

   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. 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 82.9%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if 0.0 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.9999999999999997e125

   1. Initial program 95.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 77.4%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{-h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right), \frac{D}{d}, 1\right)}} \]
   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. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if 3.9999999999999997e125 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 65.8

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

   8. Applied rewrites 65.8%

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

Alternative 3: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-22}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{d}, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\left(h \cdot M\_m\right) \cdot \left(-0.5 \cdot D\_m\right)}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) 1e-22)
     (* w0 (sqrt (fma (* (/ h l) (/ (* (* D_m M_m) -0.5) d)) t_0 1.0)))
     (* w0 (sqrt (fma t_0 (/ (* (* h M_m) (* -0.5 D_m)) (* l 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 t_0 = ((0.5 / d) * M_m) * D_m;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= 1e-22) {
		tmp = w0 * sqrt(fma(((h / l) * (((D_m * M_m) * -0.5) / d)), t_0, 1.0));
	} else {
		tmp = w0 * sqrt(fma(t_0, (((h * M_m) * (-0.5 * D_m)) / (l * 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)
	t_0 = Float64(Float64(Float64(0.5 / d) * M_m) * D_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 1e-22)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h / l) * Float64(Float64(Float64(D_m * M_m) * -0.5) / d)), t_0, 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(h * M_m) * Float64(-0.5 * D_m)) / Float64(l * 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_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, 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], 1e-22], N[(w0 * N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(-0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < 1e-22

   1. Initial program 88.3%

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

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 88.3%

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

   if 1e-22 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      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 91.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 83.4

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

   6. Applied rewrites 83.4%

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

   7. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.4

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

   9. Applied rewrites 75.4%

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

   11. Applied rewrites 84.0%

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

Alternative 4: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d}, \left(\left(0.25 \cdot M\_m\right) \cdot D\_m\right) \cdot \left(\frac{h}{\left(-d\right) \cdot \ell} \cdot M\_m\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\left(h \cdot \frac{\frac{M\_m}{d}}{\ell}\right) \cdot \frac{M\_m}{d}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+29)
   (*
    w0
    (sqrt
     (fma (/ D_m d) (* (* (* 0.25 M_m) D_m) (* (/ h (* (- d) l)) M_m)) 1.0)))
   (*
    w0
    (fma (* (* (* h (/ (/ M_m d) l)) (/ 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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+29) {
		tmp = w0 * sqrt(fma((D_m / d), (((0.25 * M_m) * D_m) * ((h / (-d * l)) * M_m)), 1.0));
	} else {
		tmp = w0 * fma((((h * ((M_m / d) / l)) * (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((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+29)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d), Float64(Float64(Float64(0.25 * M_m) * D_m) * Float64(Float64(h / Float64(Float64(-d) * l)) * M_m)), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(h * Float64(Float64(M_m / d) / l)) * 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[(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+29], N[(w0 * N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(0.25 * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(h / N[((-d) * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(h * N[(N[(M$95$m / d), $MachinePrecision] / l), $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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e29

   1. Initial program 63.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 65.3

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

   6. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 58.7

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

   8. Applied rewrites 58.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 58.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

   if -5.0000000000000001e29 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.6%

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

   3. Taylor expanded in 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 61.0%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 66.0%

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

   11. Applied rewrites 88.7%

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

Alternative 5: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(M\_m, \left(\frac{h}{\left(-d\right) \cdot \ell} \cdot D\_m\right) \cdot \left(\left(0.25 \cdot M\_m\right) \cdot \frac{D\_m}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\left(h \cdot \frac{\frac{M\_m}{d}}{\ell}\right) \cdot \frac{M\_m}{d}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+29)
   (*
    w0
    (sqrt
     (fma M_m (* (* (/ h (* (- d) l)) D_m) (* (* 0.25 M_m) (/ D_m d))) 1.0)))
   (*
    w0
    (fma (* (* (* h (/ (/ M_m d) l)) (/ 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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+29) {
		tmp = w0 * sqrt(fma(M_m, (((h / (-d * l)) * D_m) * ((0.25 * M_m) * (D_m / d))), 1.0));
	} else {
		tmp = w0 * fma((((h * ((M_m / d) / l)) * (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((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+29)
		tmp = Float64(w0 * sqrt(fma(M_m, Float64(Float64(Float64(h / Float64(Float64(-d) * l)) * D_m) * Float64(Float64(0.25 * M_m) * Float64(D_m / d))), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(h * Float64(Float64(M_m / d) / l)) * 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[(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+29], N[(w0 * N[Sqrt[N[(M$95$m * N[(N[(N[(h / N[((-d) * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(h * N[(N[(M$95$m / d), $MachinePrecision] / l), $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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e29

   1. Initial program 63.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 65.3

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

   6. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 58.7

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

   8. Applied rewrites 58.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 61.2

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

   10. Applied rewrites 61.2%

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

   if -5.0000000000000001e29 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.6%

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

   3. Taylor expanded in 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 61.0%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 66.0%

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

   11. Applied rewrites 88.7%

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

Alternative 6: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+77}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\left(h \cdot \frac{\frac{M\_m}{d}}{\ell}\right) \cdot \frac{M\_m}{d}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+77)
   (* w0 (sqrt (* (* (* (* (/ h (* d d)) M_m) (/ M_m l)) (* -0.25 D_m)) D_m)))
   (*
    w0
    (fma (* (* (* h (/ (/ M_m d) l)) (/ 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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+77) {
		tmp = w0 * sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D_m)) * D_m));
	} else {
		tmp = w0 * fma((((h * ((M_m / d) / l)) * (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((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+77)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * Float64(-0.25 * D_m)) * D_m)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(h * Float64(Float64(M_m / d) / l)) * 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[(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+77], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(h * N[(N[(M$95$m / d), $MachinePrecision] / l), $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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999997e77

   1. Initial program 61.9%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   7. Applied rewrites 51.9%

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

   if -1.99999999999999997e77 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.9%

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

   3. Taylor expanded in 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 59.7%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 64.6%

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

   11. Applied rewrites 86.9%

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

Alternative 7: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \left(-0.5 \cdot \frac{D\_m}{d}\right) \cdot \frac{h \cdot M\_m}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\left(\left(D\_m \cdot 0.5\right) \cdot h\right) \cdot M\_m}{\left(-d\right) \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m)))
   (if (<= (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) 4e+125)
     (* w0 (sqrt (fma t_0 (* (* -0.5 (/ D_m d)) (/ (* h M_m) l)) 1.0)))
     (* w0 (sqrt (fma t_0 (/ (* (* (* D_m 0.5) h) M_m) (* (- 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 t_0 = ((0.5 / d) * M_m) * D_m;
	double tmp;
	if (pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 4e+125) {
		tmp = w0 * sqrt(fma(t_0, ((-0.5 * (D_m / d)) * ((h * M_m) / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(t_0, ((((D_m * 0.5) * h) * M_m) / (-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)
	t_0 = Float64(Float64(Float64(0.5 / d) * M_m) * D_m)
	tmp = 0.0
	if ((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 4e+125)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(-0.5 * Float64(D_m / d)) * Float64(Float64(h * M_m) / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(Float64(D_m * 0.5) * h) * M_m) / Float64(Float64(-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_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e+125], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(-0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] / N[((-d) * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.9999999999999997e125

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if 3.9999999999999997e125 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 65.8

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

   8. Applied rewrites 65.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \left(-0.5 \cdot \frac{D}{d}\right) \cdot \frac{h \cdot M}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{\left(\left(D \cdot 0.5\right) \cdot h\right) \cdot M}{\left(-d\right) \cdot \ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+269}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{\frac{\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m}{\ell}}{d \cdot d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 1e+269)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (/ (/ (* (* (* D_m M_m) M_m) D_m) 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 ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 1e+269) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), (((((D_m * M_m) * M_m) * D_m) / 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 (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 1e+269)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * D_m) / 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[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], 1e+269], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] / N[(d * d), $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))) < 1e269

   1. Initial program 99.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.4%

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

   if 1e269 < (-.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 37.5%

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

   3. Taylor expanded in 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 50.6%

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

   6. Taylor expanded in D around inf

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

   8. Applied rewrites 59.1%

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

Alternative 9: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+228}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125}{\ell}, \frac{\left(\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0\right) \cdot D\_m\right) \cdot D\_m}{d \cdot d}, w0\right)\\ \end{array} \end{array} \]

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))) 5e+228)
   (* w0 1.0)
   (fma (/ -0.125 l) (/ (* (* (* (* (* M_m M_m) h) w0) D_m) D_m) (* d d)) w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+228) {
		tmp = w0 * 1.0;
	} else {
		tmp = fma((-0.125 / l), ((((((M_m * M_m) * h) * w0) * D_m) * D_m) / (d * d)), w0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+228)
		tmp = Float64(w0 * 1.0);
	else
		tmp = fma(Float64(-0.125 / l), Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) * D_m) * D_m) / Float64(d * d)), w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[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], 5e+228], N[(w0 * 1.0), $MachinePrecision], N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 5e228

   1. Initial program 99.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.9%

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

   if 5e228 < (-.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 38.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 50.3%

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

   11. Applied rewrites 51.6%

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

Alternative 10: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 10^{+104}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot \frac{D\_m}{d}\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, \frac{\left(\left(D\_m \cdot 0.5\right) \cdot h\right) \cdot M\_m}{\left(-d\right) \cdot \ell}, 1\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) 1e+104)
   (*
    w0
    (sqrt (fma (* (* -0.25 (/ D_m d)) (/ (* (* M_m M_m) h) l)) (/ D_m d) 1.0)))
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d) M_m) D_m)
      (/ (* (* (* D_m 0.5) h) M_m) (* (- 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 (pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 1e+104) {
		tmp = w0 * sqrt(fma(((-0.25 * (D_m / d)) * (((M_m * M_m) * h) / l)), (D_m / d), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((0.5 / d) * M_m) * D_m), ((((D_m * 0.5) * h) * M_m) / (-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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 1e+104)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(D_m / d)) * Float64(Float64(Float64(M_m * M_m) * h) / l)), Float64(D_m / d), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d) * M_m) * D_m), Float64(Float64(Float64(Float64(D_m * 0.5) * h) * M_m) / Float64(Float64(-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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+104], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision] / N[((-d) * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 1e104

   1. Initial program 90.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 81.6%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 1e104 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 57.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 64.6

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

   8. Applied rewrites 64.6%

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

Alternative 11: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot \frac{D\_m}{d}\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, \frac{\left(h \cdot M\_m\right) \cdot \left(-0.5 \cdot D\_m\right)}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) 4e+125)
   (*
    w0
    (sqrt (fma (* (* -0.25 (/ D_m d)) (/ (* (* M_m M_m) h) l)) (/ D_m d) 1.0)))
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d) M_m) D_m)
      (/ (* (* h M_m) (* -0.5 D_m)) (* l 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 (pow(((M_m * D_m) / (2.0 * d)), 2.0) <= 4e+125) {
		tmp = w0 * sqrt(fma(((-0.25 * (D_m / d)) * (((M_m * M_m) * h) / l)), (D_m / d), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((0.5 / d) * M_m) * D_m), (((h * M_m) * (-0.5 * D_m)) / (l * 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 ((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 4e+125)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(D_m / d)) * Float64(Float64(Float64(M_m * M_m) * h) / l)), Float64(D_m / d), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d) * M_m) * D_m), Float64(Float64(Float64(h * M_m) * Float64(-0.5 * D_m)) / Float64(l * 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[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e+125], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(-0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.9999999999999997e125

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. 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 80.8%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if 3.9999999999999997e125 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.2

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

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.1

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

   9. Applied rewrites 62.1%

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

   11. Applied rewrites 65.9%

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

Alternative 12: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+228}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \frac{h \cdot M\_m}{\left(\ell \cdot d\right) \cdot d}, 1\right)\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 5e+228)
   (* w0 1.0)
   (*
    w0
    (fma (* (* D_m D_m) -0.125) (* M_m (/ (* h M_m) (* (* 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 ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+228) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * ((h * M_m) / ((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 (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+228)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(h * M_m) / Float64(Float64(l * 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[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], 5e+228], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $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))) < 5e228

   1. Initial program 99.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.9%

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

   if 5e228 < (-.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 38.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 50.0%

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

   7. Applied rewrites 52.4%

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

   9. Applied rewrites 53.7%

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

   11. Applied rewrites 55.2%

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

Alternative 13: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.8 \cdot 10^{-274}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m, \frac{\left(h \cdot M\_m\right) \cdot \left(-0.5 \cdot D\_m\right)}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

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 1.8e-274)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d) M_m) D_m)
      (/ (* (* h M_m) (* -0.5 D_m)) (* l 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 <= 1.8e-274) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma((((0.5 / d) * M_m) * D_m), (((h * M_m) * (-0.5 * D_m)) / (l * 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 <= 1.8e-274)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d) * M_m) * D_m), Float64(Float64(Float64(h * M_m) * Float64(-0.5 * D_m)) / Float64(l * 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, 1.8e-274], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(-0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.79999999999999991e-274

   1. Initial program 81.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 67.1%

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

   if 1.79999999999999991e-274 < M

   1. Initial program 78.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 90.9

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

   6. Applied rewrites 90.9%

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

   7. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.2

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

   9. Applied rewrites 84.2%

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

   11. Applied rewrites 84.1%

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

Alternative 14: 78.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(\frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0}{d} \cdot D\_m\right) \cdot \frac{D\_m}{d}}{\ell}, w0\right)\\ \end{array} \end{array} \]

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 2.3e-160)
   (* w0 1.0)
   (fma -0.125 (/ (* (* (/ (* (* (* M_m M_m) h) w0) d) D_m) (/ D_m d)) l) w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.3e-160) {
		tmp = w0 * 1.0;
	} else {
		tmp = fma(-0.125, (((((((M_m * M_m) * h) * w0) / d) * D_m) * (D_m / d)) / l), w0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2.3e-160)
		tmp = Float64(w0 * 1.0);
	else
		tmp = fma(-0.125, Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) / d) * D_m) * Float64(D_m / d)) / l), w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2.3e-160], N[(w0 * 1.0), $MachinePrecision], N[(-0.125 * N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.29999999999999985e-160

   1. Initial program 82.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 70.7%

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

   if 2.29999999999999985e-160 < M

   1. Initial program 75.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 89.2

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

   6. Applied rewrites 89.2%

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

   7. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 60.9%

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

   11. Applied rewrites 62.9%

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

Alternative 15: 81.7% accurate, 2.4× speedup

Math

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

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 (fma (* (* (* h (/ (/ M_m d) l)) (/ 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) {
	return w0 * fma((((h * ((M_m / d) / l)) * (M_m / d)) * D_m), (-0.125 * D_m), 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 * fma(Float64(Float64(Float64(h * Float64(Float64(M_m / d) / l)) * Float64(M_m / d)) * D_m), Float64(-0.125 * D_m), 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[(N[(N[(N[(h * N[(N[(M$95$m / d), $MachinePrecision] / l), $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. Initial program 80.1%

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

3. Taylor expanded in 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 54.4%

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

7. Applied rewrites 55.2%

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

9. Applied rewrites 58.0%

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

11. Applied rewrites 75.7%

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

Alternative 16: 78.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \mathsf{fma}\left(\frac{-0.125}{\ell}, \frac{\left(\frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot w0}{d} \cdot D\_m\right) \cdot D\_m}{d}, w0\right) \end{array} \]

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
 (fma (/ -0.125 l) (/ (* (* (/ (* (* (* M_m M_m) h) w0) d) D_m) D_m) d) w0))

C

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

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return fma(Float64(-0.125 / l), Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) / d) * D_m) * D_m) / d), w0)
end

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 87.2

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

6. Applied rewrites 87.2%

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

7. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. lower-fma.f64 N/A

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

9. Applied rewrites 59.2%

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

11. Applied rewrites 68.1%

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

Alternative 17: 68.8% 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 80.1%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.7%

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

Reproduce

herbie shell --seed 2024322 
(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))))))