Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 88.5%

Time: 11.5s

Alternatives: 11

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)}{\ell}, \left(h \cdot \left(M\_m \cdot \frac{D\_m}{d\_m}\right)\right) \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \frac{-0.5 \cdot M\_m}{d\_m}, \left(\frac{h}{\ell} \cdot D\_m\right) \cdot \left(\frac{0.5}{d\_m} \cdot M\_m\right), 1\right)}\\ \end{array} \end{array} \]

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 D_m) (* 2.0 d_m)) 5e+16)
   (*
    w0
    (sqrt
     (fma
      (/ (* D_m (* M_m (/ 0.5 d_m))) l)
      (* (* h (* M_m (/ D_m d_m))) -0.5)
      1.0)))
   (*
    w0
    (sqrt
     (fma
      (* D_m (/ (* -0.5 M_m) d_m))
      (* (* (/ h l) D_m) (* (/ 0.5 d_m) M_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 * D_m) / (2.0 * d_m)) <= 5e+16) {
		tmp = w0 * sqrt(fma(((D_m * (M_m * (0.5 / d_m))) / l), ((h * (M_m * (D_m / d_m))) * -0.5), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m * ((-0.5 * M_m) / d_m)), (((h / l) * D_m) * ((0.5 / d_m) * M_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(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+16)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d_m))) / l), Float64(Float64(h * Float64(M_m * Float64(D_m / d_m))) * -0.5), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(Float64(-0.5 * M_m) / d_m)), Float64(Float64(Float64(h / l) * D_m) * Float64(Float64(0.5 / d_m) * M_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[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+16], N[(w0 * N[Sqrt[N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M$95$m * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(N[(-0.5 * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5e16

   1. Initial program 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 93.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.5

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

   9. Applied rewrites 89.5%

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

   11. Applied rewrites 93.0%

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

   if 5e16 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   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-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.6

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

   6. Applied rewrites 76.6%

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

Alternative 2: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{-0.5}{d\_m}\right), \frac{\left(h \cdot D\_m\right) \cdot \left(M\_m \cdot 0.5\right)}{\ell \cdot d\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -20000000.0)
   (*
    w0
    (sqrt
     (fma
      (* D_m (* M_m (/ -0.5 d_m)))
      (/ (* (* h D_m) (* M_m 0.5)) (* l d_m))
      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)) <= -20000000.0) {
		tmp = w0 * sqrt(fma((D_m * (M_m * (-0.5 / d_m))), (((h * D_m) * (M_m * 0.5)) / (l * d_m)), 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)) <= -20000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m * Float64(-0.5 / d_m))), Float64(Float64(Float64(h * D_m) * Float64(M_m * 0.5)) / Float64(l * d_m)), 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], -20000000.0], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(-0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * D$95$m), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $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)) < -2e7

   1. Initial program 66.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 62.9

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

   6. Applied rewrites 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 65.4

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

   8. Applied rewrites 65.4%

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

   if -2e7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 92.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.3%

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

Alternative 3: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -20000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(M\_m \cdot \left(M\_m \cdot \frac{D\_m}{d\_m}\right)\right) \cdot \frac{D\_m}{\ell \cdot d\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -20000000.0)
   (*
    w0
    (sqrt
     (fma (* h -0.25) (* (* M_m (* M_m (/ D_m d_m))) (/ D_m (* l d_m))) 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)) <= -20000000.0) {
		tmp = w0 * sqrt(fma((h * -0.25), ((M_m * (M_m * (D_m / d_m))) * (D_m / (l * d_m))), 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)) <= -20000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(M_m * Float64(M_m * Float64(D_m / d_m))) * Float64(D_m / Float64(l * d_m))), 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], -20000000.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(M$95$m * N[(M$95$m * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(l * d$95$m), $MachinePrecision]), $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)) < -2e7

   1. Initial program 66.5%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   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 44.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 65.4%

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

   if -2e7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 92.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.3%

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

Alternative 4: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+92}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, M\_m \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -5e+92)
   (*
    w0
    (sqrt
     (fma (* h -0.25) (* M_m (* (* D_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)) <= -5e+92) {
		tmp = w0 * sqrt(fma((h * -0.25), (M_m * ((D_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)) <= -5e+92)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(M_m * Float64(Float64(D_m * M_m) * Float64(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], -5e+92], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $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)) < -5.00000000000000022e92

   1. Initial program 65.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 45.8%

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

   7. Applied rewrites 54.6%

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

   if -5.00000000000000022e92 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 92.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.3%

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

Alternative 5: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M\_m}{d\_m} \cdot \frac{D\_m \cdot M\_m}{d\_m}\right) \cdot D\_m}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \left(M\_m \cdot \frac{-0.5}{d\_m}\right), \frac{\left(h \cdot D\_m\right) \cdot \left(M\_m \cdot 0.5\right)}{\ell \cdot d\_m}, 1\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) 1e+304)
   (*
    w0
    (sqrt
     (fma (* h -0.25) (/ (* (* (/ M_m d_m) (/ (* D_m M_m) d_m)) D_m) l) 1.0)))
   (*
    w0
    (sqrt
     (fma
      (* D_m (* M_m (/ -0.5 d_m)))
      (/ (* (* h D_m) (* M_m 0.5)) (* 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 (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 1e+304) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m / d_m) * ((D_m * M_m) / d_m)) * D_m) / l), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m * (M_m * (-0.5 / d_m))), (((h * D_m) * (M_m * 0.5)) / (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(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) <= 1e+304)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m / d_m) * Float64(Float64(D_m * M_m) / d_m)) * D_m) / l), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(M_m * Float64(-0.5 / d_m))), Float64(Float64(Float64(h * D_m) * Float64(M_m * 0.5)) / 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+304], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(M$95$m * N[(-0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * D$95$m), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 9.9999999999999994e303

   1. Initial program 89.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   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 64.5%

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

   7. Applied rewrites 91.8%

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

   if 9.9999999999999994e303 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 60.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. div-inv N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 66.3

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

   6. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 68.3

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

   8. Applied rewrites 68.3%

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

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\left(M\_m \cdot \left(M\_m \cdot \left(\frac{w0}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot h\right)\right)\right) \cdot D\_m, -0.125 \cdot D\_m, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -5e+92)
   (fma
    (* (* M_m (* M_m (* (/ w0 (* (* d_m d_m) l)) h))) D_m)
    (* -0.125 D_m)
    w0)
   (* 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)) <= -5e+92) {
		tmp = fma(((M_m * (M_m * ((w0 / ((d_m * d_m) * l)) * h))) * D_m), (-0.125 * D_m), w0);
	} 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)) <= -5e+92)
		tmp = fma(Float64(Float64(M_m * Float64(M_m * Float64(Float64(w0 / Float64(Float64(d_m * d_m) * l)) * h))) * D_m), Float64(-0.125 * D_m), w0);
	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], -5e+92], N[(N[(N[(M$95$m * N[(M$95$m * N[(N[(w0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + w0), $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)) < -5.00000000000000022e92

   1. Initial program 65.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 39.6%

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 51.4%

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

   if -5.00000000000000022e92 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 92.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.3%

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

Alternative 7: 76.8% accurate, 0.8× speedup

Math

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

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)) (- INFINITY))
   (fma
    (* (* D_m D_m) -0.125)
    (* (* (* M_m M_m) h) (/ w0 (* (* l d_m) d_m)))
    w0)
   (* 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)) <= -((double) INFINITY)) {
		tmp = fma(((D_m * D_m) * -0.125), (((M_m * M_m) * h) * (w0 / ((l * d_m) * d_m))), w0);
	} 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)) <= Float64(-Inf))
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(M_m * M_m) * h) * Float64(w0 / Float64(Float64(l * d_m) * d_m))), w0);
	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], (-Infinity)], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(w0 / N[(N[(l * d$95$m), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $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)) < -inf.0

   1. Initial program 58.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 48.1%

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

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

   1. Initial program 92.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.8%

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

Alternative 8: 76.4% accurate, 0.8× speedup

Math

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

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)) (- INFINITY))
   (fma
    (* (* D_m D_m) -0.125)
    (* (* (* M_m M_m) h) (/ w0 (* (* d_m d_m) l)))
    w0)
   (* 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)) <= -((double) INFINITY)) {
		tmp = fma(((D_m * D_m) * -0.125), (((M_m * M_m) * h) * (w0 / ((d_m * d_m) * l))), w0);
	} 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)) <= Float64(-Inf))
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(M_m * M_m) * h) * Float64(w0 / Float64(Float64(d_m * d_m) * l))), w0);
	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], (-Infinity)], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(w0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $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)) < -inf.0

   1. Initial program 58.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 46.2%

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

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

   1. Initial program 92.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.8%

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

Alternative 9: 87.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+16}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M\_m}{d\_m} \cdot \frac{D\_m \cdot M\_m}{d\_m}\right) \cdot D\_m}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(D\_m \cdot \frac{-0.5 \cdot M\_m}{d\_m}, \left(\frac{h}{\ell} \cdot D\_m\right) \cdot \left(\frac{0.5}{d\_m} \cdot M\_m\right), 1\right)}\\ \end{array} \end{array} \]

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 D_m) (* 2.0 d_m)) 5e+16)
   (*
    w0
    (sqrt
     (fma (* h -0.25) (/ (* (* (/ M_m d_m) (/ (* D_m M_m) d_m)) D_m) l) 1.0)))
   (*
    w0
    (sqrt
     (fma
      (* D_m (/ (* -0.5 M_m) d_m))
      (* (* (/ h l) D_m) (* (/ 0.5 d_m) M_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 * D_m) / (2.0 * d_m)) <= 5e+16) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m / d_m) * ((D_m * M_m) / d_m)) * D_m) / l), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m * ((-0.5 * M_m) / d_m)), (((h / l) * D_m) * ((0.5 / d_m) * M_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(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+16)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m / d_m) * Float64(Float64(D_m * M_m) / d_m)) * D_m) / l), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m * Float64(Float64(-0.5 * M_m) / d_m)), Float64(Float64(Float64(h / l) * D_m) * Float64(Float64(0.5 / d_m) * M_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[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+16], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m * N[(N[(-0.5 * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.5 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5e16

   1. Initial program 87.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. rgt-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 65.5%

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

   7. Applied rewrites 90.6%

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

   if 5e16 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   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-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.6

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

   6. Applied rewrites 76.6%

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

Alternative 10: 89.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\\ w0 \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{\ell}, \left(-h\right) \cdot t\_0, 1\right)} \end{array} \end{array} \]

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 (* D_m (* M_m (/ 0.5 d_m)))))
   (* w0 (sqrt (fma (/ t_0 l) (* (- h) t_0) 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 = D_m * (M_m * (0.5 / d_m));
	return w0 * sqrt(fma((t_0 / l), (-h * t_0), 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)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d_m)))
	return Float64(w0 * sqrt(fma(Float64(t_0 / l), Float64(Float64(-h) * t_0), 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_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(N[(t$95$0 / l), $MachinePrecision] * N[((-h) * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 84.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. div-inv N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 90.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

6. Applied rewrites 90.6%

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

7. Final simplification 90.6%

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

Alternative 11: 67.1% 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 84.1%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.6%

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

Reproduce

herbie shell --seed 2024307 
(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))))))