Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.9%

Time: 14.9s

Alternatives: 16

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% 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: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;D\_m \cdot M\_m \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{d} \cdot h}{\ell}, D\_m \cdot t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0, \left(\frac{D\_m}{d} \cdot \left(\frac{-0.5}{\ell} \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= (* D_m M_m) 2e+186)
     (* (sqrt (fma (/ (* (/ (* (* D_m M_m) -0.5) d) h) l) (* D_m t_0) 1.0)) w0)
     (*
      (sqrt (fma t_0 (* (* (/ D_m d) (* (/ -0.5 l) M_m)) (* h D_m)) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if ((D_m * M_m) <= 2e+186) {
		tmp = sqrt(fma((((((D_m * M_m) * -0.5) / d) * h) / l), (D_m * t_0), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(t_0, (((D_m / d) * ((-0.5 / l) * M_m)) * (h * D_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (Float64(D_m * M_m) <= 2e+186)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * -0.5) / d) * h) / l), Float64(D_m * t_0), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(t_0, Float64(Float64(Float64(D_m / d) * Float64(Float64(-0.5 / l) * M_m)) * Float64(h * D_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D$95$m * M$95$m), $MachinePrecision], 2e+186], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.5 / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.99999999999999996e186

   1. Initial program 84.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-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 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if 1.99999999999999996e186 < (*.f64 M D)

   1. Initial program 41.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-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 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 43.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 43.1

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 52.5

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

   8. Applied rewrites 52.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.5

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

   10. Applied rewrites 57.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{\left(D \cdot M\right) \cdot -0.5}{d} \cdot h}{\ell}, D \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(M \cdot \frac{0.5}{d}, \left(\frac{D}{d} \cdot \left(\frac{-0.5}{\ell} \cdot M\right)\right) \cdot \left(h \cdot D\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq 10^{-59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{d}, D\_m \cdot t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0, \left(\frac{D\_m}{d} \cdot \left(\frac{-0.5}{\ell} \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) 1e-59)
     (* (sqrt (fma (* (/ h l) (/ (* (* D_m M_m) -0.5) d)) (* D_m t_0) 1.0)) w0)
     (*
      (sqrt (fma t_0 (* (* (/ D_m d) (* (/ -0.5 l) M_m)) (* h D_m)) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= 1e-59) {
		tmp = sqrt(fma(((h / l) * (((D_m * M_m) * -0.5) / d)), (D_m * t_0), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(t_0, (((D_m / d) * ((-0.5 / l) * M_m)) * (h * D_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= 1e-59)
		tmp = Float64(sqrt(fma(Float64(Float64(h / l) * Float64(Float64(Float64(D_m * M_m) * -0.5) / d)), Float64(D_m * t_0), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(t_0, Float64(Float64(Float64(D_m / d) * Float64(Float64(-0.5 / l) * M_m)) * Float64(h * D_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e-59], N[(N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.5 / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1e-59

   1. Initial program 89.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 88.7%

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

   if 1e-59 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 14.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 25.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.3

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

   6. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.3

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

   10. Applied rewrites 85.9%

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

4. Final simplification 88.4%

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

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq 10^{-59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{-0.5}{d} \cdot D\_m\right) \cdot M\_m\right) \cdot \frac{h}{\ell}, D\_m \cdot t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0, \left(\frac{D\_m}{d} \cdot \left(\frac{-0.5}{\ell} \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 d))))
   (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) 1e-59)
     (* (sqrt (fma (* (* (* (/ -0.5 d) D_m) M_m) (/ h l)) (* D_m t_0) 1.0)) w0)
     (*
      (sqrt (fma t_0 (* (* (/ D_m d) (* (/ -0.5 l) M_m)) (* h D_m)) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = M_m * (0.5 / d);
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= 1e-59) {
		tmp = sqrt(fma(((((-0.5 / d) * D_m) * M_m) * (h / l)), (D_m * t_0), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(t_0, (((D_m / d) * ((-0.5 / l) * M_m)) * (h * D_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(M_m * Float64(0.5 / d))
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= 1e-59)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.5 / d) * D_m) * M_m) * Float64(h / l)), Float64(D_m * t_0), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(t_0, Float64(Float64(Float64(D_m / d) * Float64(Float64(-0.5 / l) * M_m)) * Float64(h * D_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e-59], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.5 / d), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.5 / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1e-59

   1. Initial program 89.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.5

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

   6. Applied rewrites 89.5%

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

   if 1e-59 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 14.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 25.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.3

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

   6. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.3

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

   10. Applied rewrites 85.9%

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

4. Final simplification 89.1%

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

Alternative 4: 86.1% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) 1e-59)
   (*
    (sqrt
     (fma (* (* (* 0.25 M_m) (* (/ D_m d) M_m)) (/ (- h) l)) (/ D_m d) 1.0))
    w0)
   (*
    (sqrt
     (fma
      (* M_m (/ 0.5 d))
      (* (* (/ D_m d) (* (/ -0.5 l) M_m)) (* h D_m))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= 1e-59) {
		tmp = sqrt(fma((((0.25 * M_m) * ((D_m / d) * M_m)) * (-h / l)), (D_m / d), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((M_m * (0.5 / d)), (((D_m / d) * ((-0.5 / l) * M_m)) * (h * D_m)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= 1e-59)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.25 * M_m) * Float64(Float64(D_m / d) * M_m)) * Float64(Float64(-h) / l)), Float64(D_m / d), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(M_m * Float64(0.5 / d)), Float64(Float64(Float64(D_m / d) * Float64(Float64(-0.5 / l) * M_m)) * Float64(h * D_m)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e-59], N[(N[Sqrt[N[(N[(N[(N[(0.25 * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.5 / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1e-59

   1. Initial program 89.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.5

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

   6. Applied rewrites 89.5%

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

   if 1e-59 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 14.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 25.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.3

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

   6. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 79.3

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

   10. Applied rewrites 85.9%

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

4. Final simplification 89.1%

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

Alternative 5: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq 1.2:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right), \frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{\ell \cdot d} \cdot h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0))) 1.2)
   (* 1.0 w0)
   (*
    (sqrt
     (fma
      (* D_m (* M_m (/ 0.5 d)))
      (* (/ (* (* D_m M_m) -0.5) (* l d)) h)
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - ((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0))) <= 1.2) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma((D_m * (M_m * (0.5 / d))), ((((D_m * M_m) * -0.5) / (l * d)) * h), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0))) <= 1.2)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(D_m * Float64(M_m * Float64(0.5 / d))), Float64(Float64(Float64(Float64(D_m * M_m) * -0.5) / Float64(l * d)) * h), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.2], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.19999999999999996

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

   if 1.19999999999999996 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. 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 66.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

   6. Applied rewrites 64.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq 1.2:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{0.5}{d}\right), \frac{\left(D \cdot M\right) \cdot -0.5}{\ell \cdot d} \cdot h, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 47.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 54.2%

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

   9. Applied rewrites 55.5%

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

4. Final simplification 83.9%

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

Alternative 7: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -5000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot D\_m}{\left(\ell \cdot d\right) \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -5000000000.0)
   (*
    (sqrt (fma (* -0.25 h) (/ (* (* (* D_m M_m) M_m) D_m) (* (* l d) d)) 1.0))
    w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -5000000000.0) {
		tmp = sqrt(fma((-0.25 * h), ((((D_m * M_m) * M_m) * D_m) / ((l * d) * d)), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -5000000000.0)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * D_m) / Float64(Float64(l * d) * d)), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5000000000.0], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e9

   1. Initial program 64.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.1%

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

   7. Applied rewrites 49.2%

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

   9. Applied rewrites 53.8%

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

   if -5e9 < (*.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

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

   5. Applied rewrites 95.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot D}{\left(\ell \cdot d\right) \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \frac{\left(\left(h \cdot M\_m\right) \cdot M\_m\right) \cdot \frac{w0}{d}}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -5e+166)
   (fma (* (* D_m D_m) -0.125) (/ (* (* (* h M_m) M_m) (/ w0 d)) (* l d)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -5e+166) {
		tmp = fma(((D_m * D_m) * -0.125), ((((h * M_m) * M_m) * (w0 / d)) / (l * d)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -5e+166)
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(Float64(h * M_m) * M_m) * Float64(w0 / d)) / Float64(l * d)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+166], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000002e166

   1. Initial program 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in M around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 36.7%

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

   9. Applied rewrites 43.2%

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

   if -5.0000000000000002e166 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{\left(\left(h \cdot M\right) \cdot M\right) \cdot \frac{w0}{d}}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{D\_m \cdot D\_m}{\ell} \cdot \left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot M\_m\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+139)
   (fma (* -0.125 w0) (* (/ (* D_m D_m) l) (* (* (/ h (* d d)) M_m) M_m)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+139) {
		tmp = fma((-0.125 * w0), (((D_m * D_m) / l) * (((h / (d * d)) * M_m) * M_m)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+139)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(D_m * D_m) / l) * Float64(Float64(Float64(h / Float64(d * d)) * M_m) * M_m)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+139], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000007e139

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in M around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 40.5%

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

   12. Applied rewrites 39.0%

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

   if -2.00000000000000007e139 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d \cdot d} \cdot M\right) \cdot M\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in M around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 39.7%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 46.3%

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

   12. Applied rewrites 46.4%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 87.6%

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

4. Final simplification 80.1%

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

Alternative 11: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{\left(\ell \cdot d\right) \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -5e+166)
   (fma (* -0.125 w0) (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* (* l d) d)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -5e+166) {
		tmp = fma((-0.125 * w0), (((((M_m * M_m) * h) * D_m) * D_m) / ((l * d) * d)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -5e+166)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(Float64(l * d) * d)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -5e+166], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000002e166

   1. Initial program 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in M around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 42.8%

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

   12. Applied rewrites 44.8%

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

   if -5.0000000000000002e166 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. frac-times N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in M around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 39.7%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 46.3%

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

   12. Applied rewrites 46.6%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 87.6%

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

4. Final simplification 80.1%

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

Alternative 13: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{M\_m}{\ell} \cdot \left(\frac{D\_m}{d} \cdot M\_m\right)\right) \cdot D\_m}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right), \frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{\ell \cdot d} \cdot h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ (* D_m M_m) (* 2.0 d)) 2e+30)
   (*
    (sqrt (fma (* -0.25 h) (/ (* (* (/ M_m l) (* (/ D_m d) M_m)) D_m) d) 1.0))
    w0)
   (*
    (sqrt
     (fma
      (* D_m (* M_m (/ 0.5 d)))
      (* (/ (* (* D_m M_m) -0.5) (* l d)) h)
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((D_m * M_m) / (2.0 * d)) <= 2e+30) {
		tmp = sqrt(fma((-0.25 * h), ((((M_m / l) * ((D_m / d) * M_m)) * D_m) / d), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((D_m * (M_m * (0.5 / d))), ((((D_m * M_m) * -0.5) / (l * d)) * h), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) <= 2e+30)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(M_m / l) * Float64(Float64(D_m / d) * M_m)) * D_m) / d), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(D_m * Float64(M_m * Float64(0.5 / d))), Float64(Float64(Float64(Float64(D_m * M_m) * -0.5) / Float64(l * d)) * h), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e+30], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2e30

   1. Initial program 86.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 89.6%

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

   if 2e30 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 57.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 64.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

   6. Applied rewrites 64.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{2 \cdot d} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(\frac{M}{\ell} \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot D}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(D \cdot \left(M \cdot \frac{0.5}{d}\right), \frac{\left(D \cdot M\right) \cdot -0.5}{\ell \cdot d} \cdot h, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 87.3%

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

5. Final simplification 87.3%

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

Alternative 15: 87.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
2. Add Preprocessing
3. 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 81.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 87.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 87.7

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 87.7

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

6. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 88.4

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

8. Applied rewrites 88.4%

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

9. Final simplification 88.4%

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

Alternative 16: 67.6% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 72.3%

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

6. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2024276 
(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))))))