Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.2%

Time: 16.8s

Alternatives: 5

Speedup: 1.7×

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: 80.7% 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: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ t_1 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-54}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h \cdot \left(t\_1 \cdot t\_1\right)}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+144}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(t\_0 \cdot \left(D\_m \cdot \frac{M\_m \cdot 0.5}{d\_m}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0\_m} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))) (t_1 (* M_m (/ (* D_m 0.5) d_m))))
   (*
    w0_s
    (if (<= t_0 1e-54)
      (* w0_m (sqrt (- 1.0 (/ (* h (* t_1 t_1)) l))))
      (if (<= t_0 4e+144)
        (* w0_m (sqrt (- 1.0 (* (* t_0 (* D_m (/ (* M_m 0.5) d_m))) (/ h l)))))
        (pow
         (*
          (sqrt w0_m)
          (pow
           (exp 0.25)
           (+
            (log (* -0.25 (/ (* h (pow (* M_m D_m) 2.0)) l)))
            (* -2.0 (log d_m)))))
         2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (t_0 <= 1e-54) {
		tmp = w0_m * sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	} else if (t_0 <= 4e+144) {
		tmp = w0_m * sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	} else {
		tmp = pow((sqrt(w0_m) * pow(exp(0.25), (log((-0.25 * ((h * pow((M_m * D_m), 2.0)) / l))) + (-2.0 * log(d_m))))), 2.0);
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * d_m) / (2.0d0 * d_m_1)
    t_1 = m_m * ((d_m * 0.5d0) / d_m_1)
    if (t_0 <= 1d-54) then
        tmp = w0_m * sqrt((1.0d0 - ((h * (t_1 * t_1)) / l)))
    else if (t_0 <= 4d+144) then
        tmp = w0_m * sqrt((1.0d0 - ((t_0 * (d_m * ((m_m * 0.5d0) / d_m_1))) * (h / l))))
    else
        tmp = (sqrt(w0_m) * (exp(0.25d0) ** (log(((-0.25d0) * ((h * ((m_m * d_m) ** 2.0d0)) / l))) + ((-2.0d0) * log(d_m_1))))) ** 2.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (t_0 <= 1e-54) {
		tmp = w0_m * Math.sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	} else if (t_0 <= 4e+144) {
		tmp = w0_m * Math.sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.pow(Math.exp(0.25), (Math.log((-0.25 * ((h * Math.pow((M_m * D_m), 2.0)) / l))) + (-2.0 * Math.log(d_m))))), 2.0);
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (2.0 * d_m)
	t_1 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if t_0 <= 1e-54:
		tmp = w0_m * math.sqrt((1.0 - ((h * (t_1 * t_1)) / l)))
	elif t_0 <= 4e+144:
		tmp = w0_m * math.sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))))
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.pow(math.exp(0.25), (math.log((-0.25 * ((h * math.pow((M_m * D_m), 2.0)) / l))) + (-2.0 * math.log(d_m))))), 2.0)
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	t_1 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (t_0 <= 1e-54)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_1 * t_1)) / l))));
	elseif (t_0 <= 4e+144)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(D_m * Float64(Float64(M_m * 0.5) / d_m))) * Float64(h / l)))));
	else
		tmp = Float64(sqrt(w0_m) * (exp(0.25) ^ Float64(log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / l))) + Float64(-2.0 * log(d_m))))) ^ 2.0;
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (2.0 * d_m);
	t_1 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if (t_0 <= 1e-54)
		tmp = w0_m * sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	elseif (t_0 <= 4e+144)
		tmp = w0_m * sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	else
		tmp = (sqrt(w0_m) * (exp(0.25) ^ (log((-0.25 * ((h * ((M_m * D_m) ^ 2.0)) / l))) + (-2.0 * log(d_m))))) ^ 2.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 1e-54], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+144], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1e-54

   1. Initial program 84.6%

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

   3. Simplified 83.6%

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

      1. unpow2 83.6%

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

      2. unpow2 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*l/ 84.6%

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

      5. associate-*r/ 84.1%

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

      6. times-frac 84.6%

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

      7. associate-*r/ 90.1%

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

      8. *-commutative 90.1%

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

      9. associate-/l* 89.8%

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

      10. div-inv 89.8%

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

      11. associate-/r* 89.8%

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

      12. metadata-eval 89.8%

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

   6. Applied egg-rr 89.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
   7. Step-by-step derivation

      1. unpow2 89.8%

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

      2. associate-*r/ 89.8%

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

      3. associate-*r/ 89.8%

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

   8. Applied egg-rr 89.8%

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

   if 1e-54 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.00000000000000009e144

   1. Initial program 99.8%

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

   3. Simplified 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. times-frac 99.8%

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

      5. unpow2 99.8%

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

      6. associate-/r* 99.8%

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

      7. frac-times 60.1%

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

      8. div-inv 60.1%

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

      9. *-commutative 60.1%

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

      10. metadata-eval 60.1%

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

      11. associate-*r* 60.1%

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

      12. *-commutative 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. times-frac 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l* 95.8%

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

      2. *-commutative 95.8%

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

      3. *-commutative 95.8%

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

   10. Simplified 95.8%

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

   if 4.00000000000000009e144 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 49.4%

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

   3. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. unpow2 46.0%

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

      3. *-commutative 46.0%

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

      4. associate-*l/ 49.4%

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

      5. associate-*r/ 49.4%

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

      6. times-frac 49.4%

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

      7. associate-*r/ 49.5%

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

      8. *-commutative 49.5%

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

      9. associate-/l* 49.5%

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

      10. div-inv 49.5%

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

      11. associate-/r* 49.5%

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

      12. metadata-eval 49.5%

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

   6. Applied egg-rr 49.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
   7. Step-by-step derivation

      1. unpow2 49.5%

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

      2. associate-*r/ 49.5%

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

      3. associate-*r/ 49.5%

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

   8. Applied egg-rr 49.5%

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

      1. add-sqr-sqrt 33.7%

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

      2. pow2 33.7%

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

   10. Applied egg-rr 33.7%

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

   11. Taylor expanded in d around 0 22.0%

       \[\leadsto {\color{blue}{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{2} \]
   12. Step-by-step derivation

       1. exp-prod 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}}\right)}^{2} \]

       2. distribute-lft-neg-in 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)} + -2 \cdot \log d\right)}\right)}^{2} \]

       3. metadata-eval 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       4. associate-*r* 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       5. *-commutative 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot {D}^{2}\right)} \cdot h}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       6. unpow2 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}\right) \cdot h}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       7. unpow2 22.0%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot h}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       8. swap-sqr 25.1%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       9. unpow2 25.1%

          \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{{\left(M \cdot D\right)}^{2}} \cdot h}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       10. *-commutative 25.1%

           \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot D\right)}^{2}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       11. *-commutative 25.1%

           \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\color{blue}{\left(D \cdot M\right)}}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

   13. Simplified 25.1%

       \[\leadsto {\color{blue}{\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ t_1 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {t\_0}^{2} \leq 4 \cdot 10^{+303}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(t\_0 \cdot \left(D\_m \cdot \frac{M\_m \cdot 0.5}{d\_m}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h \cdot \left(t\_1 \cdot t\_1\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))) (t_1 (* M_m (/ (* D_m 0.5) d_m))))
   (*
    w0_s
    (if (<= (- 1.0 (* (/ h l) (pow t_0 2.0))) 4e+303)
      (* w0_m (sqrt (- 1.0 (* (* t_0 (* D_m (/ (* M_m 0.5) d_m))) (/ h l)))))
      (* w0_m (sqrt (- 1.0 (/ (* h (* t_1 t_1)) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((1.0 - ((h / l) * pow(t_0, 2.0))) <= 4e+303) {
		tmp = w0_m * sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	} else {
		tmp = w0_m * sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * d_m) / (2.0d0 * d_m_1)
    t_1 = m_m * ((d_m * 0.5d0) / d_m_1)
    if ((1.0d0 - ((h / l) * (t_0 ** 2.0d0))) <= 4d+303) then
        tmp = w0_m * sqrt((1.0d0 - ((t_0 * (d_m * ((m_m * 0.5d0) / d_m_1))) * (h / l))))
    else
        tmp = w0_m * sqrt((1.0d0 - ((h * (t_1 * t_1)) / l)))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if ((1.0 - ((h / l) * Math.pow(t_0, 2.0))) <= 4e+303) {
		tmp = w0_m * Math.sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (2.0 * d_m)
	t_1 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if (1.0 - ((h / l) * math.pow(t_0, 2.0))) <= 4e+303:
		tmp = w0_m * math.sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))))
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h * (t_1 * t_1)) / l)))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	t_1 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (t_0 ^ 2.0))) <= 4e+303)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(D_m * Float64(Float64(M_m * 0.5) / d_m))) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_1 * t_1)) / l))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (2.0 * d_m);
	t_1 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if ((1.0 - ((h / l) * (t_0 ^ 2.0))) <= 4e+303)
		tmp = w0_m * sqrt((1.0 - ((t_0 * (D_m * ((M_m * 0.5) / d_m))) * (h / l))));
	else
		tmp = w0_m * sqrt((1.0 - ((h * (t_1 * t_1)) / l)));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+303], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 4e303

   1. Initial program 99.9%

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

   3. Simplified 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 98.8%

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

      4. times-frac 99.9%

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

      5. unpow2 99.9%

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

      6. associate-/r* 99.9%

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

      7. frac-times 81.8%

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

      8. div-inv 81.8%

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

      9. *-commutative 81.8%

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

      10. metadata-eval 81.8%

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

      11. associate-*r* 81.8%

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

      12. *-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. times-frac 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-/l* 97.2%

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

      2. *-commutative 97.2%

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

      3. *-commutative 97.2%

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

   10. Simplified 97.2%

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

   if 4e303 < (-.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 41.7%

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

   3. Simplified 43.0%

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

      1. unpow2 43.0%

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

      2. unpow2 43.0%

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

      3. *-commutative 43.0%

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

      4. associate-*l/ 41.7%

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

      5. associate-*r/ 43.0%

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

      6. times-frac 41.7%

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

      7. associate-*r/ 58.7%

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

      8. *-commutative 58.7%

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

      9. associate-/l* 59.1%

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

      10. div-inv 59.1%

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

      11. associate-/r* 59.1%

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

      12. metadata-eval 59.1%

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

   6. Applied egg-rr 59.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
   7. Step-by-step derivation

      1. unpow2 59.1%

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

      2. associate-*r/ 59.1%

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

      3. associate-*r/ 59.1%

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

   8. Applied egg-rr 59.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+303}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-128}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m} \cdot \frac{0.5}{d\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* M_m D_m) 5e-128)
    w0_m
    (*
     w0_m
     (sqrt
      (-
       1.0
       (*
        (* M_m D_m)
        (* (/ h l) (* (/ (* M_m (* D_m 0.5)) d_m) (/ 0.5 d_m))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 5e-128) {
		tmp = w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((M_m * D_m) * ((h / l) * (((M_m * (D_m * 0.5)) / d_m) * (0.5 / d_m))))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((m_m * d_m) <= 5d-128) then
        tmp = w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - ((m_m * d_m) * ((h / l) * (((m_m * (d_m * 0.5d0)) / d_m_1) * (0.5d0 / d_m_1))))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 5e-128) {
		tmp = w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((M_m * D_m) * ((h / l) * (((M_m * (D_m * 0.5)) / d_m) * (0.5 / d_m))))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if (M_m * D_m) <= 5e-128:
		tmp = w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((M_m * D_m) * ((h / l) * (((M_m * (D_m * 0.5)) / d_m) * (0.5 / d_m))))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-128)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(M_m * D_m) * Float64(Float64(h / l) * Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) * Float64(0.5 / d_m)))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((M_m * D_m) <= 5e-128)
		tmp = w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((M_m * D_m) * ((h / l) * (((M_m * (D_m * 0.5)) / d_m) * (0.5 / d_m))))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-128], w0$95$m, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000001e-128

   1. Initial program 84.4%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in D around 0 76.2%

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

   if 5.0000000000000001e-128 < (*.f64 M D)

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-*l/ 76.2%

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

      5. associate-*r/ 77.7%

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

      6. times-frac 76.2%

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

      7. associate-*r/ 80.6%

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

      8. *-commutative 80.6%

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

      9. associate-/l* 82.0%

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

      10. div-inv 82.0%

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

      11. associate-/r* 82.0%

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

      12. metadata-eval 82.0%

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

   6. Applied egg-rr 82.0%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
   7. Step-by-step derivation

      1. unpow2 82.0%

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

      2. associate-*r/ 82.0%

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

      3. associate-*r/ 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*r/ 82.0%

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

      3. associate-*r* 80.6%

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

      4. *-commutative 80.6%

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

      5. associate-*r/ 80.6%

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

      6. associate-*r* 79.6%

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

      7. associate-*r/ 76.3%

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

      8. associate-*l* 79.1%

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

      9. *-commutative 79.1%

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

      10. *-commutative 79.1%

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

      11. associate-*r/ 79.1%

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

      12. associate-*r/ 79.1%

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

   10. Applied egg-rr 79.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-128}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot \frac{0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 3.6 \cdot 10^{-125}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(w0\_m \cdot h\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}, w0\_m\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* M_m D_m) 3.6e-125)
    w0_m
    (fma
     -0.125
     (/ (* (* (* M_m D_m) (* M_m D_m)) (* w0_m h)) (* l (* d_m d_m)))
     w0_m))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 3.6e-125) {
		tmp = w0_m;
	} else {
		tmp = fma(-0.125, ((((M_m * D_m) * (M_m * D_m)) * (w0_m * h)) / (l * (d_m * d_m))), w0_m);
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 3.6e-125)
		tmp = w0_m;
	else
		tmp = fma(-0.125, Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(w0_m * h)) / Float64(l * Float64(d_m * d_m))), w0_m);
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 3.6e-125], w0$95$m, N[(-0.125 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.6000000000000002e-125

   1. Initial program 84.4%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in D around 0 76.2%

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

   if 3.6000000000000002e-125 < (*.f64 M D)

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in D around 0 50.2%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative 50.2%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. fma-define 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

      3. associate-*r* 50.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell}, w0\right) \]

      4. unpow2 50.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      5. unpow2 50.2%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      6. swap-sqr 61.4%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      7. unpow2 61.4%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]
   8. Step-by-step derivation

      1. unpow2 61.4%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      2. *-commutative 61.4%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(D \cdot M\right)\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

      3. *-commutative 61.4%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(\left(M \cdot D\right) \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]

   9. Applied egg-rr 61.4%

      \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right) \]
   10. Step-by-step derivation

       1. unpow2 61.4%

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 3.6 \cdot 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot w0\_m \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m) :precision binary64 (* w0_s w0_m))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * w0_m
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	return w0_s * w0_m

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	return Float64(w0_s * w0_m)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = w0_s * w0_m;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Simplified 80.7%

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

5. Taylor expanded in D around 0 68.2%

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

Reproduce

herbie shell --seed 2024129 
(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))))))