Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 86.9%

Time: 17.8s

Alternatives: 16

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: 81.7% 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: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D\_m}{d} \cdot M\right) \cdot -0.5\right) \cdot \frac{h}{\ell}, D\_m \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{-0.25 \cdot M}{\ell} \cdot M\right) \cdot \frac{D\_m}{d}, \frac{D\_m}{d} \cdot h, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M) (* 2.0 d)) 2.0)) 4e-44)
   (*
    (sqrt
     (fma (* (* (* (/ D_m d) M) -0.5) (/ h l)) (* D_m (* M (/ 0.5 d))) 1.0))
    w0)
   (*
    (sqrt (fma (* (* (/ (* -0.25 M) l) M) (/ D_m d)) (* (/ D_m d) h) 1.0))
    w0)))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M) / (2.0 * d)), 2.0)) <= 4e-44) {
		tmp = sqrt(fma(((((D_m / d) * M) * -0.5) * (h / l)), (D_m * (M * (0.5 / d))), 1.0)) * w0;
	} else {
		tmp = sqrt(fma(((((-0.25 * M) / l) * M) * (D_m / d)), ((D_m / d) * h), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0)) <= 4e-44)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m / d) * M) * -0.5) * Float64(h / l)), Float64(D_m * Float64(M * Float64(0.5 / d))), 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.25 * M) / l) * M) * Float64(D_m / d)), Float64(Float64(D_m / d) * h), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 4e-44], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.25 * M), $MachinePrecision] / l), $MachinePrecision] * M), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 3.99999999999999981e-44

   1. Initial program 89.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 90.5%

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

   if 3.99999999999999981e-44 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 4.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 84.7%

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

   6. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.1

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

   8. Applied rewrites 96.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot -0.5\right) \cdot \frac{h}{\ell}, D \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{-0.25 \cdot M}{\ell} \cdot M\right) \cdot \frac{D}{d}, \frac{D}{d} \cdot h, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (/ h l) (pow (/ (* D_m M) (* 2.0 d)) 2.0))) 1.0)
   (* 1.0 w0)
   (*
    (sqrt (fma (- D_m) (/ (* (* (* D_m M) h) M) (* (* (* 4.0 l) d) d)) 1.0))
    w0)))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - ((h / l) * pow(((D_m * M) / (2.0 * d)), 2.0))) <= 1.0) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma(-D_m, ((((D_m * M) * h) * M) / (((4.0 * l) * d) * d)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0))) <= 1.0)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(-D_m), Float64(Float64(Float64(Float64(D_m * M) * h) * M) / Float64(Float64(Float64(4.0 * l) * d) * d)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[((-D$95$m) * N[(N[(N[(N[(D$95$m * M), $MachinePrecision] * h), $MachinePrecision] * M), $MachinePrecision] / N[(N[(N[(4.0 * l), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*r/ N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.0

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 54.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. associate-*r* N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 58.3%

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

4. Final simplification 83.6%

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

Reproduce

herbie shell --seed 2024234 
(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))))))