Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 87.6%

Time: 12.9s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d\_m \leq 2 \cdot 10^{-51}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{0.5}{d\_m}, M\_m \cdot \frac{D\_m \cdot \left(\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d\_m} \cdot h\right)}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell} \cdot h\right) \cdot -0.5\right) \cdot \frac{D\_m}{d\_m}, \left(\frac{0.5}{d\_m} \cdot D\_m\right) \cdot M\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* 2.0 d_m) 2e-51)
   (*
    w0
    (sqrt
     (fma
      (/ 0.5 d_m)
      (* M_m (/ (* D_m (* (/ (* -0.5 (* D_m M_m)) d_m) h)) l))
      1.0)))
   (*
    (sqrt
     (fma
      (* (* (* (/ M_m l) h) -0.5) (/ D_m d_m))
      (* (* (/ 0.5 d_m) D_m) M_m)
      1.0))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((2.0 * d_m) <= 2e-51) {
		tmp = w0 * sqrt(fma((0.5 / d_m), (M_m * ((D_m * (((-0.5 * (D_m * M_m)) / d_m) * h)) / l)), 1.0));
	} else {
		tmp = sqrt(fma(((((M_m / l) * h) * -0.5) * (D_m / d_m)), (((0.5 / d_m) * D_m) * M_m), 1.0)) * w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(2.0 * d_m) <= 2e-51)
		tmp = Float64(w0 * sqrt(fma(Float64(0.5 / d_m), Float64(M_m * Float64(Float64(D_m * Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d_m) * h)) / l)), 1.0)));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(M_m / l) * h) * -0.5) * Float64(D_m / d_m)), Float64(Float64(Float64(0.5 / d_m) * D_m) * M_m), 1.0)) * w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(2.0 * d$95$m), $MachinePrecision], 2e-51], N[(w0 * N[Sqrt[N[(N[(0.5 / d$95$m), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * h), $MachinePrecision] * -0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 2e-51

   1. Initial program 78.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 84.5%

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

   6. Applied rewrites 87.2%

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

   if 2e-51 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 90.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 95.4%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.1

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

   9. Applied rewrites 89.5%

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

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+237} \lor \neg \left(t\_0 \leq -2 \cdot 10^{-6}\right):\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(-0.125 \cdot D\_m\right) \cdot \left(\frac{h \cdot \frac{M\_m}{d\_m}}{\ell} \cdot \frac{M\_m}{d\_m}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\_m\right) \cdot \frac{h \cdot \left(M\_m \cdot D\_m\right)}{d\_m \cdot d\_m}, \frac{M\_m}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (or (<= t_0 -5e+237) (not (<= t_0 -2e-6)))
     (*
      w0
      (fma D_m (* (* -0.125 D_m) (* (/ (* h (/ M_m d_m)) l) (/ M_m d_m))) 1.0))
     (*
      w0
      (sqrt
       (fma
        (* (* -0.25 D_m) (/ (* h (* M_m D_m)) (* d_m d_m)))
        (/ M_m l)
        1.0))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if ((t_0 <= -5e+237) || !(t_0 <= -2e-6)) {
		tmp = w0 * fma(D_m, ((-0.125 * D_m) * (((h * (M_m / d_m)) / l) * (M_m / d_m))), 1.0);
	} else {
		tmp = w0 * sqrt(fma(((-0.25 * D_m) * ((h * (M_m * D_m)) / (d_m * d_m))), (M_m / l), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if ((t_0 <= -5e+237) || !(t_0 <= -2e-6))
		tmp = Float64(w0 * fma(D_m, Float64(Float64(-0.125 * D_m) * Float64(Float64(Float64(h * Float64(M_m / d_m)) / l) * Float64(M_m / d_m))), 1.0));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * D_m) * Float64(Float64(h * Float64(M_m * D_m)) / Float64(d_m * d_m))), Float64(M_m / l), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+237], N[Not[LessEqual[t$95$0, -2e-6]], $MachinePrecision]], N[(w0 * N[(D$95$m * N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000002e237 or -1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 62.5%

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

   9. Applied rewrites 85.1%

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

   if -5.0000000000000002e237 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999991e-6

   1. Initial program 99.2%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 32.0

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

   5. Applied rewrites 32.0%

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

   7. Applied rewrites 46.1%

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

   9. Applied rewrites 53.4%

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

   11. Applied rewrites 61.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+237} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-6}\right):\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, \left(-0.125 \cdot D\right) \cdot \left(\frac{h \cdot \frac{M}{d}}{\ell} \cdot \frac{M}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\right) \cdot \frac{h \cdot \left(M \cdot D\right)}{d \cdot d}, \frac{M}{\ell}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+156}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{\left(\left(h \cdot \frac{M\_m}{d\_m}\right) \cdot M\_m\right) \cdot \left(-0.125 \cdot D\_m\right)}{\ell \cdot d\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 1.0)
     (* w0 1.0)
     (if (<= t_0 2e+156)
       (*
        w0
        (sqrt
         (fma
          (* h -0.25)
          (/ (* (* (* M_m M_m) D_m) D_m) (* (* d_m d_m) l))
          1.0)))
       (*
        w0
        (fma
         D_m
         (/ (* (* (* h (/ M_m d_m)) M_m) (* -0.125 D_m)) (* l d_m))
         1.0))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = 1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 2e+156) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) * D_m) * D_m) / ((d_m * d_m) * l)), 1.0));
	} else {
		tmp = w0 * fma(D_m, ((((h * (M_m / d_m)) * M_m) * (-0.125 * D_m)) / (l * d_m)), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 2e+156)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) / Float64(Float64(d_m * d_m) * l)), 1.0)));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(Float64(h * Float64(M_m / d_m)) * M_m) * Float64(-0.125 * D_m)) / Float64(l * d_m)), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+156], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

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

   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))) < 2e156

   1. Initial program 99.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   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 62.4%

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

   if 2e156 < (-.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 48.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 42.5%

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 57.6%

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

   11. Applied rewrites 55.3%

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

Alternative 4: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, \left(\frac{h}{\ell} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{-0.5}{d\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.25 \cdot D\_m\right) \cdot \left(D\_m \cdot \left(\frac{h \cdot \frac{M\_m}{d\_m}}{d\_m} \cdot \frac{M\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
      2e+129)
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d_m) M_m) D_m)
      (* (* (/ h l) (* D_m M_m)) (/ -0.5 d_m))
      1.0)))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* 0.25 D_m) (* D_m (* (/ (* h (/ M_m d_m)) d_m) (/ M_m l)))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 2e+129) {
		tmp = w0 * sqrt(fma((((0.5 / d_m) * M_m) * D_m), (((h / l) * (D_m * M_m)) * (-0.5 / d_m)), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * D_m) * (D_m * (((h * (M_m / d_m)) / d_m) * (M_m / l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))) <= 2e+129)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), Float64(Float64(Float64(h / l) * Float64(D_m * M_m)) * Float64(-0.5 / d_m)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * D_m) * Float64(D_m * Float64(Float64(Float64(h * Float64(M_m / d_m)) / d_m) * Float64(M_m / l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e+129], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 2e129

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

   6. Applied rewrites 98.3%

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

   if 2e129 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 42.6%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 64.4%

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

Alternative 5: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-183}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot -0.5}{d\_m}, \left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\_m\right) \cdot \frac{\left(h \cdot \frac{M\_m}{d\_m}\right) \cdot D\_m}{d\_m}, \frac{M\_m}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) 5e-183)
   (*
    w0
    (sqrt
     (fma
      (* (/ h l) (/ (* (* D_m M_m) -0.5) d_m))
      (* (* (/ 0.5 d_m) M_m) D_m)
      1.0)))
   (*
    w0
    (sqrt
     (fma
      (* (* -0.25 D_m) (/ (* (* h (/ M_m d_m)) D_m) d_m))
      (/ M_m l)
      1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 5e-183) {
		tmp = w0 * sqrt(fma(((h / l) * (((D_m * M_m) * -0.5) / d_m)), (((0.5 / d_m) * M_m) * D_m), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((-0.25 * D_m) * (((h * (M_m / d_m)) * D_m) / d_m)), (M_m / l), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= 5e-183)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h / l) * Float64(Float64(Float64(D_m * M_m) * -0.5) / d_m)), Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * D_m) * Float64(Float64(Float64(h * Float64(M_m / d_m)) * D_m) / d_m)), Float64(M_m / l), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-183], N[(w0 * N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.0000000000000002e-183

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 89.4%

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

   if 5.0000000000000002e-183 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 34.5%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 63.6%

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

   9. Applied rewrites 66.5%

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

   11. Applied rewrites 83.0%

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

Alternative 6: 85.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

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

   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 56.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 69.8%

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

Alternative 7: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, h \cdot \frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{\ell \cdot d\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \left(-0.125 \cdot D\_m\right) \cdot \left(\frac{h \cdot \frac{M\_m}{d\_m}}{\ell} \cdot \frac{M\_m}{d\_m}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e-6)
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d_m) M_m) D_m)
      (* h (/ (* -0.5 (* D_m M_m)) (* l d_m)))
      1.0)))
   (*
    w0
    (fma D_m (* (* -0.125 D_m) (* (/ (* h (/ M_m d_m)) l) (/ M_m d_m))) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e-6) {
		tmp = w0 * sqrt(fma((((0.5 / d_m) * M_m) * D_m), (h * ((-0.5 * (D_m * M_m)) / (l * d_m))), 1.0));
	} else {
		tmp = w0 * fma(D_m, ((-0.125 * D_m) * (((h * (M_m / d_m)) / l) * (M_m / d_m))), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e-6)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), Float64(h * Float64(Float64(-0.5 * Float64(D_m * M_m)) / Float64(l * d_m))), 1.0)));
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(-0.125 * D_m) * Float64(Float64(Float64(h * Float64(M_m / d_m)) / l) * Float64(M_m / d_m))), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-6], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h * N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999991e-6

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

   6. Applied rewrites 69.2%

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

   if -1.99999999999999991e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 95.1%

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

Alternative 8: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e-8)
   (*
    w0
    (sqrt
     (fma (* h -0.25) (/ (* (* (* D_m D_m) M_m) M_m) (* (* d_m d_m) l)) 1.0)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e-8) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((D_m * D_m) * M_m) * M_m) / ((d_m * d_m) * l)), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e-8)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * M_m) / Float64(Float64(d_m * d_m) * l)), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-8], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e-8

   1. Initial program 68.4%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 42.0

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in h around inf

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

   10. Applied rewrites 48.3%

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

   if -1e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.2%

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

Alternative 9: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, D\_m \cdot \left(M\_m \cdot \frac{M\_m \cdot D\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e-8)
   (*
    w0
    (fma (* h -0.125) (* D_m (* M_m (/ (* M_m D_m) (* (* d_m d_m) l)))) 1.0))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e-8) {
		tmp = w0 * fma((h * -0.125), (D_m * (M_m * ((M_m * D_m) / ((d_m * d_m) * l)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e-8)
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(D_m * Float64(M_m * Float64(Float64(M_m * D_m) / Float64(Float64(d_m * d_m) * l)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-8], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(D$95$m * N[(M$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e-8

   1. Initial program 68.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 39.7%

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

   10. Applied rewrites 39.8%

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

   12. Applied rewrites 44.0%

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

   if -1e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.2%

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

Alternative 10: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+234}:\\ \;\;\;\;w0 \cdot \frac{\left(-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d\_m \cdot d\_m\right) \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+234)
   (* w0 (/ (* (* -0.125 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d_m d_m) l)))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+234) {
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d_m * d_m) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+234)) then
        tmp = w0 * ((((-0.125d0) * (((m_m * d_m) * m_m) * d_m)) * h) / ((d_m_1 * d_m_1) * l))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+234) {
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d_m * d_m) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+234:
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d_m * d_m) * l))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+234)
		tmp = Float64(w0 * Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d_m * d_m) * l)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+234)
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d_m * d_m) * l));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+234], N[(w0 * N[(N[(N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000004e234

   1. Initial program 59.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 46.4%

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

   10. Applied rewrites 46.5%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 51.1%

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

   if -2.00000000000000004e234 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 88.3%

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

Alternative 11: 85.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-166}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\frac{0.5}{d\_m} \cdot M\_m\right) \cdot D\_m, \left(-0.5 \cdot \frac{D\_m}{d\_m}\right) \cdot \frac{h \cdot M\_m}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot D\_m\right) \cdot \frac{\left(h \cdot \frac{M\_m}{d\_m}\right) \cdot D\_m}{d\_m}, \frac{M\_m}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -4e-166)
   (*
    w0
    (sqrt
     (fma
      (* (* (/ 0.5 d_m) M_m) D_m)
      (* (* -0.5 (/ D_m d_m)) (/ (* h M_m) l))
      1.0)))
   (*
    w0
    (sqrt
     (fma
      (* (* -0.25 D_m) (/ (* (* h (/ M_m d_m)) D_m) d_m))
      (/ M_m l)
      1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -4e-166) {
		tmp = w0 * sqrt(fma((((0.5 / d_m) * M_m) * D_m), ((-0.5 * (D_m / d_m)) * ((h * M_m) / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma(((-0.25 * D_m) * (((h * (M_m / d_m)) * D_m) / d_m)), (M_m / l), 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -4e-166)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(0.5 / d_m) * M_m) * D_m), Float64(Float64(-0.5 * Float64(D_m / d_m)) * Float64(Float64(h * M_m) / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * D_m) * Float64(Float64(Float64(h * Float64(M_m / d_m)) * D_m) / d_m)), Float64(M_m / l), 1.0)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -4e-166], N[(w0 * N[Sqrt[N[(N[(N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(-0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.00000000000000016e-166

   1. Initial program 78.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in M around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -4.00000000000000016e-166 < (/.f64 h l)

   1. Initial program 86.3%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 82.7%

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

   11. Applied rewrites 90.6%

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

Alternative 12: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d\_m \leq 10^{-150}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{\left(\left(\left(-0.125 \cdot D\_m\right) \cdot h\right) \cdot \frac{\frac{M\_m}{d\_m}}{\ell}\right) \cdot M\_m}{d\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.25 \cdot D\_m\right) \cdot \left(D\_m \cdot \left(\left(h \cdot \frac{M\_m}{d\_m \cdot d\_m}\right) \cdot \frac{M\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* 2.0 d_m) 1e-150)
   (*
    w0
    (fma D_m (/ (* (* (* (* -0.125 D_m) h) (/ (/ M_m d_m) l)) M_m) d_m) 1.0))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* 0.25 D_m) (* D_m (* (* h (/ M_m (* d_m d_m))) (/ M_m l)))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((2.0 * d_m) <= 1e-150) {
		tmp = w0 * fma(D_m, (((((-0.125 * D_m) * h) * ((M_m / d_m) / l)) * M_m) / d_m), 1.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * D_m) * (D_m * ((h * (M_m / (d_m * d_m))) * (M_m / l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(2.0 * d_m) <= 1e-150)
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(Float64(Float64(-0.125 * D_m) * h) * Float64(Float64(M_m / d_m) / l)) * M_m) / d_m), 1.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * D_m) * Float64(D_m * Float64(Float64(h * Float64(M_m / Float64(d_m * d_m))) * Float64(M_m / l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(2.0 * d$95$m), $MachinePrecision], 1e-150], N[(w0 * N[(D$95$m * N[(N[(N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[(h * N[(M$95$m / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 1.00000000000000001e-150

   1. Initial program 78.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 49.8%

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

   7. Applied rewrites 56.0%

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

   9. Applied rewrites 73.4%

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

   11. Applied rewrites 70.1%

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

   if 1.00000000000000001e-150 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 88.7%

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

   3. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 83.4%

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

   9. Applied rewrites 89.0%

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

Alternative 13: 88.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 88.0%

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

Alternative 14: 79.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, \frac{\left(\left(h \cdot \frac{M\_m}{d\_m}\right) \cdot M\_m\right) \cdot \left(-0.125 \cdot D\_m\right)}{\ell \cdot d\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 1.8e-51)
   (* w0 1.0)
   (*
    w0
    (fma D_m (/ (* (* (* h (/ M_m d_m)) M_m) (* -0.125 D_m)) (* l d_m)) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.8e-51) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(D_m, ((((h * (M_m / d_m)) * M_m) * (-0.125 * D_m)) / (l * d_m)), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.8e-51)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(D_m, Float64(Float64(Float64(Float64(h * Float64(M_m / d_m)) * M_m) * Float64(-0.125 * D_m)) / Float64(l * d_m)), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 1.8e-51], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(D$95$m * N[(N[(N[(N[(h * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.8e-51

   1. Initial program 81.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 70.7%

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

   if 1.8e-51 < D

   1. Initial program 86.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 77.8%

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

   11. Applied rewrites 76.2%

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

Alternative 15: 78.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.4 \cdot 10^{-157}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, D\_m \cdot \left(\frac{D\_m}{d\_m} \cdot \frac{M\_m \cdot M\_m}{\ell \cdot d\_m}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 5.4e-157)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (* D_m (* (/ D_m d_m) (/ (* M_m M_m) (* l d_m)))) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 5.4e-157) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), (D_m * ((D_m / d_m) * ((M_m * M_m) / (l * d_m)))), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 5.4e-157)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(D_m * Float64(Float64(D_m / d_m) * Float64(Float64(M_m * M_m) / Float64(l * d_m)))), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 5.4e-157], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(D$95$m * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.4e-157

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 72.9%

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

   if 5.4e-157 < M

   1. Initial program 81.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 57.6%

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

   10. Applied rewrites 59.8%

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

   12. Applied rewrites 65.1%

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

Alternative 16: 80.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * fma(D_m, Float64(Float64(-0.125 * D_m) * Float64(Float64(Float64(h * Float64(M_m / d_m)) / l) * Float64(M_m / d_m))), 1.0))
end

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 55.5%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 79.8%

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

Alternative 17: 67.5% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    code = w0 * 1.0d0
end function

Java

d_m = Math.abs(d);
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_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * 1.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2024319 
(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))))))