Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 89.1%

Time: 14.7s

Alternatives: 18

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: 80.5% 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: 89.1% accurate, 1.8× speedup

Math

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

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

Derivation

1. Initial program 81.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. lift-/.f64 N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 87.6

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

6. Applied rewrites 87.6%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. lower-*.f64 87.6

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 87.6

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

8. Applied rewrites 87.6%

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

9. Final simplification 87.6%

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

Alternative 2: 85.9% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 48.9%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 53.1%

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

   9. Applied rewrites 67.3%

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

4. Final simplification 88.2%

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

Alternative 3: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M\_m \cdot D\_m}{\ell \cdot d} \cdot \frac{M\_m \cdot D\_m}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (*
    (sqrt (fma (* -0.25 h) (* (/ (* M_m D_m) (* l d)) (/ (* M_m D_m) d)) 1.0))
    w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = sqrt(fma((-0.25 * h), (((M_m * D_m) / (l * d)) * ((M_m * D_m) / d)), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(M_m * D_m) / Float64(l * d)) * Float64(Float64(M_m * D_m) / d)), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.050000000000000003

   1. Initial program 66.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 58.0%

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

   if -0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M \cdot D}{\ell \cdot d} \cdot \frac{M \cdot D}{d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M\_m \cdot D\_m}{\ell} \cdot \frac{M\_m \cdot D\_m}{d \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (*
    (sqrt (fma (* -0.25 h) (* (/ (* M_m D_m) l) (/ (* M_m D_m) (* d d))) 1.0))
    w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = sqrt(fma((-0.25 * h), (((M_m * D_m) / l) * ((M_m * D_m) / (d * d))), 1.0)) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(M_m * D_m) / l) * Float64(Float64(M_m * D_m) / Float64(d * d))), 1.0)) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.050000000000000003

   1. Initial program 66.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 57.9%

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

   if -0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M \cdot D}{\ell} \cdot \frac{M \cdot D}{d \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(M\_m \cdot D\_m\right) \cdot -0.5, \frac{\left(\left(M\_m \cdot 0.5\right) \cdot h\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (*
    (sqrt
     (fma
      (* (* M_m D_m) -0.5)
      (/ (* (* (* M_m 0.5) h) D_m) (* (* d d) l))
      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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = sqrt(fma(((M_m * D_m) * -0.5), ((((M_m * 0.5) * h) * D_m) / ((d * d) * l)), 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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(sqrt(fma(Float64(Float64(M_m * D_m) * -0.5), Float64(Float64(Float64(Float64(M_m * 0.5) * h) * D_m) / Float64(Float64(d * d) * l)), 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.050000000000000003

   1. Initial program 66.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 59.3

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

   6. Applied rewrites 59.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 53.2%

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

   if -0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(M \cdot D\right) \cdot -0.5, \frac{\left(\left(M \cdot 0.5\right) \cdot h\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M\_m \cdot D\_m}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M\_m \cdot D\_m\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (*
    (sqrt (fma (* -0.25 h) (* (/ (* M_m D_m) (* (* d d) l)) (* M_m 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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = sqrt(fma((-0.25 * h), (((M_m * D_m) / ((d * d) * l)) * (M_m * 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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(M_m * D_m) / Float64(Float64(d * d) * l)) * Float64(M_m * 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.050000000000000003

   1. Initial program 66.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 52.4%

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

   if -0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{M \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot D\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(M\_m \cdot D\_m\right) \cdot M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot D\_m, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -0.05)
   (*
    (sqrt (fma (* -0.25 h) (* (/ (* (* M_m D_m) M_m) (* (* d d) l)) 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 ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -0.05) {
		tmp = sqrt(fma((-0.25 * h), ((((M_m * D_m) * M_m) / ((d * d) * l)) * 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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -0.05)
		tmp = Float64(sqrt(fma(Float64(-0.25 * h), Float64(Float64(Float64(Float64(M_m * D_m) * M_m) / Float64(Float64(d * d) * l)) * 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.050000000000000003

   1. Initial program 66.7%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 50.6%

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

   if -0.050000000000000003 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.05:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25 \cdot h, \frac{\left(M \cdot D\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot D, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(\frac{w0}{d} \cdot \left(h \cdot M\_m\right)\right) \cdot M\_m}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+189)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* (/ w0 d) (* h M_m)) M_m) (* l d)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+189) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((w0 / d) * (h * M_m)) * M_m) / (l * d)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+189)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(w0 / d) * Float64(h * M_m)) * M_m) / Float64(l * d)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+189], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(w0 / d), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000004e189

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 60.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\frac{\frac{d}{M}}{D} \cdot 2}}} \]

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 41.8%

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

   9. Applied rewrites 44.0%

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

   11. Applied rewrites 45.9%

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

   if -5.0000000000000004e189 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 91.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(\frac{w0}{d} \cdot \left(h \cdot M\right)\right) \cdot M}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(\frac{w0}{d \cdot d} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+149)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* (/ w0 (* d d)) M_m) (* h M_m)) l) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+149) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((w0 / (d * d)) * M_m) * (h * M_m)) / l), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+149)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(w0 / Float64(d * d)) * M_m) * Float64(h * M_m)) / l), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+149], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(w0 / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $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)) < -1.00000000000000005e149

   1. Initial program 62.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 62.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\frac{\frac{d}{M}}{D} \cdot 2}}} \]

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 41.5%

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

   9. Applied rewrites 43.5%

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

   if -1.00000000000000005e149 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(\frac{w0}{d \cdot d} \cdot M\right) \cdot \left(h \cdot M\right)}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+220)
   (fma (* -0.125 w0) (/ (* (* (* D_m D_m) M_m) (* h M_m)) (* (* d d) l)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+220) {
		tmp = fma((-0.125 * w0), ((((D_m * D_m) * M_m) * (h * M_m)) / ((d * d) * l)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+220)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * Float64(h * M_m)) / Float64(Float64(d * d) * l)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+220], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e220

   1. Initial program 58.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 58.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\frac{\frac{d}{M}}{D} \cdot 2}}} \]

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 49.3%

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

   12. Applied rewrites 45.5%

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

   if -2e220 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot \left(h \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell} \cdot D\_m, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+220)
   (fma (* -0.125 w0) (* (/ (* (* (* M_m M_m) h) D_m) (* (* d d) l)) D_m) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+220) {
		tmp = fma((-0.125 * w0), (((((M_m * M_m) * h) * D_m) / ((d * d) * l)) * D_m), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+220)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / Float64(Float64(d * d) * l)) * D_m), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+220], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e220

   1. Initial program 58.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 58.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)}{\frac{\frac{d}{M}}{D} \cdot 2}}} \]

   5. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 49.3%

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

   12. Applied rewrites 49.4%

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

   if -2e220 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot D, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.7% accurate, 1.5× speedup

Math

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

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 D_m) (* 2.0 d)) 5e+30)
   (*
    (sqrt (fma (/ D_m d) (* (* -0.25 h) (* (/ (* M_m D_m) d) (/ M_m l))) 1.0))
    w0)
   (*
    (sqrt
     (fma
      (* (/ (* -0.5 M_m) d) D_m)
      (/ (* (* M_m 0.5) (* h D_m)) (* 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 * D_m) / (2.0 * d)) <= 5e+30) {
		tmp = sqrt(fma((D_m / d), ((-0.25 * h) * (((M_m * D_m) / d) * (M_m / l))), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((-0.5 * M_m) / d) * D_m), (((M_m * 0.5) * (h * D_m)) / (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 (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) <= 5e+30)
		tmp = Float64(sqrt(fma(Float64(D_m / d), Float64(Float64(-0.25 * h) * Float64(Float64(Float64(M_m * D_m) / d) * Float64(M_m / l))), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(-0.5 * M_m) / d) * D_m), Float64(Float64(Float64(M_m * 0.5) * Float64(h * D_m)) / 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[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+30], N[(N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(-0.5 * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.9999999999999998e30

   1. Initial program 85.3%

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

   3. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 88.7%

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

   if 4.9999999999999998e30 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 59.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 71.0

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

   6. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

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

4. Final simplification 86.2%

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

Alternative 13: 76.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot M\_m}{d} \cdot D\_m, \frac{\left(M\_m \cdot 0.5\right) \cdot \left(h \cdot D\_m\right)}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 D_m) (* 2.0 d)) 2e-34)
   (* 1.0 w0)
   (*
    (sqrt
     (fma
      (* (/ (* -0.5 M_m) d) D_m)
      (/ (* (* M_m 0.5) (* h D_m)) (* 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 * D_m) / (2.0 * d)) <= 2e-34) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma((((-0.5 * M_m) / d) * D_m), (((M_m * 0.5) * (h * D_m)) / (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 (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) <= 2e-34)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(-0.5 * M_m) / d) * D_m), Float64(Float64(Float64(M_m * 0.5) * Float64(h * D_m)) / 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[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e-34], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(-0.5 * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.99999999999999986e-34

   1. Initial program 84.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 78.6%

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

   if 1.99999999999999986e-34 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 68.5%

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 72.7

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

   6. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot M}{d} \cdot D, \frac{\left(M \cdot 0.5\right) \cdot \left(h \cdot D\right)}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\_m\\ \mathbf{if}\;2 \cdot d \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d}, \left(\frac{h}{\ell} \cdot D\_m\right) \cdot t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0 \cdot D\_m, \frac{h \cdot M\_m}{\ell} \cdot \left(\frac{D\_m}{d} \cdot -0.5\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M_m)))
   (if (<= (* 2.0 d) 2e-112)
     (* (sqrt (fma (/ (* (* M_m D_m) -0.5) d) (* (* (/ h l) D_m) t_0) 1.0)) w0)
     (*
      (sqrt (fma (* t_0 D_m) (* (/ (* h M_m) l) (* (/ D_m d) -0.5)) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * M_m;
	double tmp;
	if ((2.0 * d) <= 2e-112) {
		tmp = sqrt(fma((((M_m * D_m) * -0.5) / d), (((h / l) * D_m) * t_0), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((t_0 * D_m), (((h * M_m) / l) * ((D_m / d) * -0.5)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * M_m)
	tmp = 0.0
	if (Float64(2.0 * d) <= 2e-112)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / d), Float64(Float64(Float64(h / l) * D_m) * t_0), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(t_0 * D_m), Float64(Float64(Float64(h * M_m) / l) * Float64(Float64(D_m / d) * -0.5)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(2.0 * d), $MachinePrecision], 2e-112], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 1.9999999999999999e-112

   1. Initial program 78.7%

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 76.3%

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

   if 1.9999999999999999e-112 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 87.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 94.5%

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

   5. Taylor expanded in h around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d}, \left(\frac{h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{h \cdot M}{\ell} \cdot \left(\frac{D}{d} \cdot -0.5\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\_m\\ \mathbf{if}\;2 \cdot d \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot -0.5}{d}, \left(\frac{D\_m}{\ell} \cdot h\right) \cdot t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_0 \cdot D\_m, \frac{h \cdot M\_m}{\ell} \cdot \left(\frac{D\_m}{d} \cdot -0.5\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M_m)))
   (if (<= (* 2.0 d) 2e-112)
     (* (sqrt (fma (/ (* (* M_m D_m) -0.5) d) (* (* (/ D_m l) h) t_0) 1.0)) w0)
     (*
      (sqrt (fma (* t_0 D_m) (* (/ (* h M_m) l) (* (/ D_m d) -0.5)) 1.0))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (0.5 / d) * M_m;
	double tmp;
	if ((2.0 * d) <= 2e-112) {
		tmp = sqrt(fma((((M_m * D_m) * -0.5) / d), (((D_m / l) * h) * t_0), 1.0)) * w0;
	} else {
		tmp = sqrt(fma((t_0 * D_m), (((h * M_m) / l) * ((D_m / d) * -0.5)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * M_m)
	tmp = 0.0
	if (Float64(2.0 * d) <= 2e-112)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(M_m * D_m) * -0.5) / d), Float64(Float64(Float64(D_m / l) * h) * t_0), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(t_0 * D_m), Float64(Float64(Float64(h * M_m) / l) * Float64(Float64(D_m / d) * -0.5)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(2.0 * d), $MachinePrecision], 2e-112], N[(N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(D$95$m / l), $MachinePrecision] * h), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 1.9999999999999999e-112

   1. Initial program 78.7%

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.5

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

   6. Applied rewrites 80.5%

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

   if 1.9999999999999999e-112 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 87.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 94.5%

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

   5. Taylor expanded in h around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{d}, \left(\frac{D}{\ell} \cdot h\right) \cdot \left(\frac{0.5}{d} \cdot M\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D, \frac{h \cdot M}{\ell} \cdot \left(\frac{D}{d} \cdot -0.5\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.1% accurate, 1.9× speedup

Math

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

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

Derivation

1. Initial program 81.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 87.6%

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

5. Final simplification 87.6%

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

Alternative 17: 88.4% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 81.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/l* N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l* N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 85.4

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

6. Applied rewrites 87.7%

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

7. Final simplification 87.7%

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

Alternative 18: 68.0% 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 81.7%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 72.7%

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

6. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024244 
(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))))))