Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 82.2% → 89.2%

Time: 7.0s

Alternatives: 14

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: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% 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 81.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 88.1%

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

5. Final simplification 88.1%

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

Alternative 2: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot \left(\frac{-D\_m}{d} \cdot \frac{h}{\ell}\right), 0.25, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot h}{d}}{d} \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+28)
   (*
    (sqrt
     (fma (* (* (* (/ D_m d) M_m) M_m) (* (/ (- D_m) d) (/ h l))) 0.25 1.0))
    w0)
   (*
    (fma (* (/ (/ (* (* (/ M_m l) M_m) h) d) d) D_m) (* -0.125 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)) <= -2e+28) {
		tmp = sqrt(fma(((((D_m / d) * M_m) * M_m) * ((-D_m / d) * (h / l))), 0.25, 1.0)) * w0;
	} else {
		tmp = fma(((((((M_m / l) * M_m) * h) / d) / d) * D_m), (-0.125 * 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)) <= -2e+28)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m / d) * M_m) * M_m) * Float64(Float64(Float64(-D_m) / d) * Float64(h / l))), 0.25, 1.0)) * w0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) / d) * D_m), Float64(-0.125 * 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], -2e+28], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[((-D$95$m) / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $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)) < -1.99999999999999992e28

   1. Initial program 66.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 68.3%

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

   6. Applied rewrites 61.0%

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

   if -1.99999999999999992e28 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 84.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot M\right) \cdot \left(\frac{-D}{d} \cdot \frac{h}{\ell}\right), 0.25, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M}{\ell} \cdot M\right) \cdot h}{d}}{d} \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(h \cdot \frac{M\_m}{d}\right) \cdot M\_m\right) \cdot D\_m\right) \cdot \left(-0.25 \cdot D\_m\right)}{\ell}}{d}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot h}{d}}{d} \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+28)
   (* (sqrt (/ (/ (* (* (* (* h (/ M_m d)) M_m) D_m) (* -0.25 D_m)) l) d)) w0)
   (*
    (fma (* (/ (/ (* (* (/ M_m l) M_m) h) d) d) D_m) (* -0.125 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)) <= -2e+28) {
		tmp = sqrt(((((((h * (M_m / d)) * M_m) * D_m) * (-0.25 * D_m)) / l) / d)) * w0;
	} else {
		tmp = fma(((((((M_m / l) * M_m) * h) / d) / d) * D_m), (-0.125 * 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)) <= -2e+28)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(h * Float64(M_m / d)) * M_m) * D_m) * Float64(-0.25 * D_m)) / l) / d)) * w0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) / d) * D_m), Float64(-0.125 * 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], -2e+28], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $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)) < -1.99999999999999992e28

   1. Initial program 66.0%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.2

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

   5. Applied rewrites 44.2%

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

   7. Applied rewrites 49.2%

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

   9. Applied rewrites 62.2%

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

   if -1.99999999999999992e28 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 84.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(h \cdot \frac{M}{d}\right) \cdot M\right) \cdot D\right) \cdot \left(-0.25 \cdot D\right)}{\ell}}{d}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M}{\ell} \cdot M\right) \cdot h}{d}}{d} \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot D\_m} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot h}{d}}{d} \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -1e+75)
   (* (sqrt (* (* (* (* (/ h (* d d)) M_m) (/ M_m l)) (* -0.25 D_m)) D_m)) w0)
   (*
    (fma (* (/ (/ (* (* (/ M_m l) M_m) h) d) d) D_m) (* -0.125 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+75) {
		tmp = sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D_m)) * D_m)) * w0;
	} else {
		tmp = fma(((((((M_m / l) * M_m) * h) / d) / d) * D_m), (-0.125 * 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+75)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * Float64(-0.25 * D_m)) * D_m)) * w0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) / d) * D_m), Float64(-0.125 * 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+75], N[(N[Sqrt[N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $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)) < -9.99999999999999927e74

   1. Initial program 65.1%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   7. Applied rewrites 55.5%

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

   if -9.99999999999999927e74 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 57.8%

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

   7. Applied rewrites 79.5%

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

   9. Applied rewrites 83.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{h}{d \cdot d} \cdot M\right) \cdot \frac{M}{\ell}\right) \cdot \left(-0.25 \cdot D\right)\right) \cdot D} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M}{\ell} \cdot M\right) \cdot h}{d}}{d} \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 0.8× speedup

Math

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

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)) (- INFINITY))
   (* (fma (* (/ (* (* h M_m) M_m) (* (* d d) l)) D_m) (* -0.125 D_m) 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)) <= -((double) INFINITY)) {
		tmp = fma(((((h * M_m) * M_m) / ((d * d) * l)) * D_m), (-0.125 * D_m), 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)) <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(Float64(h * M_m) * M_m) / Float64(Float64(d * d) * l)) * D_m), Float64(-0.125 * D_m), 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], (-Infinity)], N[(N[(N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $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 59.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 49.3%

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

   7. Applied rewrites 57.9%

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

   9. Applied rewrites 53.0%

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

   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 88.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.7%

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

4. Final simplification 81.4%

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

Alternative 6: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \frac{M\_m \cdot M\_m}{\left(\ell \cdot d\right) \cdot d} \cdot h, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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)) -2e+252)
   (* (fma (* (* D_m D_m) -0.125) (* (/ (* M_m M_m) (* (* l d) d)) h) 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)) <= -2e+252) {
		tmp = fma(((D_m * D_m) * -0.125), (((M_m * M_m) / ((l * d) * d)) * h), 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)) <= -2e+252)
		tmp = Float64(fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(M_m * M_m) / Float64(Float64(l * d) * d)) * h), 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], -2e+252], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $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)) < -2.0000000000000002e252

   1. Initial program 61.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 47.1%

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

   9. Applied rewrites 50.3%

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

   if -2.0000000000000002e252 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{M \cdot M}{\left(\ell \cdot d\right) \cdot d} \cdot h, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.8× speedup

Math

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

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)) (- INFINITY))
   (* (fma (* (* D_m D_m) -0.125) (* (/ (* M_m M_m) (* (* d d) l)) h) 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)) <= -((double) INFINITY)) {
		tmp = fma(((D_m * D_m) * -0.125), (((M_m * M_m) / ((d * d) * l)) * h), 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)) <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(M_m * M_m) / Float64(Float64(d * d) * l)) * h), 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], (-Infinity)], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $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 59.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 49.3%

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

   7. Applied rewrites 49.3%

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

   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 88.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.7%

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

4. Final simplification 80.5%

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

Alternative 8: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d} \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot h}{d} \cdot \frac{D\_m}{d}, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \frac{h}{d}}{\ell} \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ (* D_m M_m) (* 2.0 d)) 4e+102)
   (* (fma (* (/ (* (* (/ M_m l) M_m) h) d) (/ D_m d)) (* -0.125 D_m) 1.0) w0)
   (*
    (fma (* (/ (* (* (/ M_m d) M_m) (/ h d)) l) D_m) (* -0.125 D_m) 1.0)
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((D_m * M_m) / (2.0 * d)) <= 4e+102) {
		tmp = fma((((((M_m / l) * M_m) * h) / d) * (D_m / d)), (-0.125 * D_m), 1.0) * w0;
	} else {
		tmp = fma((((((M_m / d) * M_m) * (h / d)) / l) * D_m), (-0.125 * D_m), 1.0) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) <= 4e+102)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) * Float64(D_m / d)), Float64(-0.125 * D_m), 1.0) * w0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64(h / d)) / l) * D_m), Float64(-0.125 * D_m), 1.0) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 4e+102], N[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 3.99999999999999991e102

   1. Initial program 84.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 67.7%

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

   11. Applied rewrites 76.8%

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

   if 3.99999999999999991e102 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 61.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 64.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{2 \cdot d} \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{M}{\ell} \cdot M\right) \cdot h}{d} \cdot \frac{D}{d}, -0.125 \cdot D, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{M}{d} \cdot M\right) \cdot \frac{h}{d}}{\ell} \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 81.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 88.1%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-/.f64 N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. lower-fma.f64 N/A

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

6. Applied rewrites 87.4%

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

7. Taylor expanded in M around 0

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

   1. times-frac N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 82.3

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

9. Applied rewrites 82.3%

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

10. Final simplification 82.3%

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

Alternative 10: 83.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot h}{d}}{d} \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot \frac{D\_m}{d}, \left(-D\_m\right) \cdot \frac{h}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 (<= M_m 9e-142)
   (* (fma (* (/ (/ (* (* (/ M_m l) M_m) h) d) d) D_m) (* -0.125 D_m) 1.0) w0)
   (*
    (sqrt
     (fma (* (* (* M_m M_m) 0.25) (/ D_m d)) (* (- D_m) (/ h (* l d))) 1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 9e-142) {
		tmp = fma(((((((M_m / l) * M_m) * h) / d) / d) * D_m), (-0.125 * D_m), 1.0) * w0;
	} else {
		tmp = sqrt(fma((((M_m * M_m) * 0.25) * (D_m / d)), (-D_m * (h / (l * d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 9e-142)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) / d) * D_m), Float64(-0.125 * D_m), 1.0) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m * M_m) * 0.25) * Float64(D_m / d)), Float64(Float64(-D_m) * Float64(h / Float64(l * d))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 9e-142], N[(N[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[((-D$95$m) * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.00000000000000037e-142

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 82.3%

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

   if 9.00000000000000037e-142 < M

   1. Initial program 68.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   4. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.5

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

   6. Applied rewrites 64.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\frac{M}{\ell} \cdot M\right) \cdot h}{d}}{d} \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \frac{D}{d}, \left(-D\right) \cdot \frac{h}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \cdot M\_m \leq 5 \cdot 10^{-82}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right) \cdot w0\\ \end{array} \end{array} \]

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) 5e-82)
   (* 1.0 w0)
   (*
    (fma (* (* (* (/ h (* d d)) M_m) (/ M_m l)) D_m) (* -0.125 D_m) 1.0)
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((D_m * M_m) <= 5e-82) {
		tmp = 1.0 * w0;
	} else {
		tmp = fma(((((h / (d * d)) * M_m) * (M_m / l)) * D_m), (-0.125 * 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(D_m * M_m) <= 5e-82)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * D_m), Float64(-0.125 * 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[(D$95$m * M$95$m), $MachinePrecision], 5e-82], N[(1.0 * w0), $MachinePrecision], N[(N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4.9999999999999998e-82

   1. Initial program 85.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 79.9%

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

   if 4.9999999999999998e-82 < (*.f64 M D)

   1. Initial program 68.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 64.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-82}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{h}{d \cdot d} \cdot M\right) \cdot \frac{M}{\ell}\right) \cdot D, -0.125 \cdot D, 1\right) \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.8% accurate, 2.4× speedup

Math

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

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
 (* (fma (* (/ (/ (* (* (/ M_m l) M_m) h) d) d) D_m) (* -0.125 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) {
	return fma(((((((M_m / l) * M_m) * h) / d) / d) * D_m), (-0.125 * D_m), 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(fma(Float64(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) / d) * D_m), Float64(-0.125 * D_m), 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[(N[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 53.5%

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

7. Applied rewrites 71.2%

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

9. Applied rewrites 74.3%

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

10. Final simplification 74.3%

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

Alternative 13: 78.6% accurate, 2.4× speedup

Math

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

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
 (* (fma (* (/ (* (* (/ M_m l) M_m) h) d) (/ D_m d)) (* -0.125 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) {
	return fma((((((M_m / l) * M_m) * h) / d) * (D_m / d)), (-0.125 * D_m), 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(fma(Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * h) / d) * Float64(D_m / d)), Float64(-0.125 * D_m), 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[(N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

3. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 53.5%

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

7. Applied rewrites 71.2%

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

9. Applied rewrites 66.9%

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

11. Applied rewrites 74.9%

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

12. Final simplification 74.9%

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

Alternative 14: 68.2% accurate, 26.2× speedup

Math

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

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 81.5%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.5%

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

6. Final simplification 69.5%

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

Reproduce

herbie shell --seed 2024332 
(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))))))