Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 88.1%

Time: 15.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% 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: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+259}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left(h \cdot \frac{{\left(M_m \cdot D_m\right)}^{2}}{\ell}\right)\right) + -2 \cdot \log d_m\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M_m}{d_m} \cdot \frac{D_m}{2}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 1 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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+259)
    (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)
    (*
     w0_m
     (sqrt (- 1.0 (/ (* h (pow (* (/ M_m d_m) (/ D_m 2.0)) 2.0)) 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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+259) {
		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);
	} else {
		tmp = w0_m * sqrt((1.0 - ((h * pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / 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) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+259)) then
        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
    else
        tmp = w0_m * sqrt((1.0d0 - ((h * (((m_m / d_m_1) * (d_m / 2.0d0)) ** 2.0d0)) / 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 tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+259) {
		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);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h * Math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / 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):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+259:
		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)
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h * math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / 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)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+259)
		tmp = Float64(sqrt(w0_m) * (exp(0.25) ^ Float64(log(Float64(-0.25 * Float64(h * Float64((Float64(M_m * D_m) ^ 2.0) / l)))) + Float64(-2.0 * log(d_m))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m / d_m) * Float64(D_m / 2.0)) ^ 2.0)) / 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)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+259)
		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;
	else
		tmp = w0_m * sqrt((1.0 - ((h * (((M_m / d_m) * (D_m / 2.0)) ^ 2.0)) / 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_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+259], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(h * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2e259

   1. Initial program 53.3%

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

   3. Simplified 53.4%

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

      1. *-commutative 53.4%

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

      2. frac-times 53.3%

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

      3. associate-*l/ 53.3%

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

      4. associate-*l/ 56.1%

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

      5. associate-*l/ 57.4%

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

      6. *-commutative 57.4%

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

      7. times-frac 56.1%

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

   5. Applied egg-rr 56.1%

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

      1. add-sqr-sqrt 29.9%

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

      2. pow2 29.9%

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

      3. *-commutative 29.9%

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

      4. sqrt-prod 29.8%

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

      5. unpow2 29.8%

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

      6. sqrt-prod 17.9%

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

      7. add-sqr-sqrt 31.2%

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

      8. div-inv 31.2%

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

      9. metadata-eval 31.2%

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

   7. Applied egg-rr 31.2%

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

      1. associate-*l* 31.2%

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

      2. *-commutative 31.2%

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

   9. Simplified 31.2%

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

       1. add-sqr-sqrt 16.0%

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

       2. pow2 16.0%

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

   11. Applied egg-rr 25.0%

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

   12. Taylor expanded in d around 0 8.9%

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

       1. exp-prod 8.9%

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

   14. Simplified 11.7%

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

   if -2e259 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Simplified 90.2%

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

      1. *-commutative 90.2%

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

      2. frac-times 90.2%

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

      3. associate-*l/ 90.8%

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

      4. associate-*l/ 97.4%

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

      5. associate-*l/ 96.9%

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

      6. *-commutative 96.9%

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

      7. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+259}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \left(h \cdot \frac{{\left(M \cdot D\right)}^{2}}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \left(\frac{M_m}{d_m} \cdot \left(D_m \cdot 0.5\right)\right) \cdot \sqrt{h}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;h \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M_m}{d_m} \cdot \frac{D_m}{2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\ell}}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 1 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) (* D_m 0.5)) (sqrt h))))
   (*
    w0_s
    (if (<= h 1.5e-295)
      (* w0_m (sqrt (- 1.0 (/ (* h (pow (* (/ M_m d_m) (/ D_m 2.0)) 2.0)) l))))
      (pow (sqrt (* w0_m (sqrt (- 1.0 (* t_0 (/ t_0 l)))))) 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) * (D_m * 0.5)) * sqrt(h);
	double tmp;
	if (h <= 1.5e-295) {
		tmp = w0_m * sqrt((1.0 - ((h * pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l)));
	} else {
		tmp = pow(sqrt((w0_m * sqrt((1.0 - (t_0 * (t_0 / l)))))), 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) :: tmp
    t_0 = ((m_m / d_m_1) * (d_m * 0.5d0)) * sqrt(h)
    if (h <= 1.5d-295) then
        tmp = w0_m * sqrt((1.0d0 - ((h * (((m_m / d_m_1) * (d_m / 2.0d0)) ** 2.0d0)) / l)))
    else
        tmp = sqrt((w0_m * sqrt((1.0d0 - (t_0 * (t_0 / l)))))) ** 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) * (D_m * 0.5)) * Math.sqrt(h);
	double tmp;
	if (h <= 1.5e-295) {
		tmp = w0_m * Math.sqrt((1.0 - ((h * Math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l)));
	} else {
		tmp = Math.pow(Math.sqrt((w0_m * Math.sqrt((1.0 - (t_0 * (t_0 / l)))))), 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) * (D_m * 0.5)) * math.sqrt(h)
	tmp = 0
	if h <= 1.5e-295:
		tmp = w0_m * math.sqrt((1.0 - ((h * math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l)))
	else:
		tmp = math.pow(math.sqrt((w0_m * math.sqrt((1.0 - (t_0 * (t_0 / l)))))), 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(Float64(M_m / d_m) * Float64(D_m * 0.5)) * sqrt(h))
	tmp = 0.0
	if (h <= 1.5e-295)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m / d_m) * Float64(D_m / 2.0)) ^ 2.0)) / l))));
	else
		tmp = sqrt(Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 / l)))))) ^ 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) * (D_m * 0.5)) * sqrt(h);
	tmp = 0.0;
	if (h <= 1.5e-295)
		tmp = w0_m * sqrt((1.0 - ((h * (((M_m / d_m) * (D_m / 2.0)) ^ 2.0)) / l)));
	else
		tmp = sqrt((w0_m * sqrt((1.0 - (t_0 * (t_0 / l)))))) ^ 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[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[h, 1.5e-295], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.49999999999999998e-295

   1. Initial program 81.2%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-times 81.2%

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

      3. associate-*l/ 82.0%

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

      4. associate-*l/ 90.4%

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

      5. associate-*l/ 90.3%

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

      6. *-commutative 90.3%

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

      7. times-frac 90.4%

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

   5. Applied egg-rr 90.4%

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

   if 1.49999999999999998e-295 < h

   1. Initial program 78.9%

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

   3. Simplified 79.7%

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

      1. *-commutative 79.7%

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

      2. frac-times 78.9%

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

      3. associate-*l/ 79.0%

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

      4. associate-*l/ 82.1%

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

      5. associate-*l/ 82.1%

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

      6. *-commutative 82.1%

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

      7. times-frac 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. add-sqr-sqrt 82.1%

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

      2. pow2 82.1%

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

      3. *-commutative 82.1%

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

      4. sqrt-prod 82.1%

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

      5. unpow2 82.1%

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

      6. sqrt-prod 61.8%

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

      7. add-sqr-sqrt 84.8%

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

      8. div-inv 84.8%

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

      9. metadata-eval 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. associate-*l* 83.3%

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

      2. *-commutative 83.3%

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

   9. Simplified 83.3%

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

       1. add-sqr-sqrt 45.2%

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

       2. pow2 45.2%

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

   11. Applied egg-rr 43.9%

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

       1. add-sqr-sqrt 43.9%

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

       2. *-un-lft-identity 43.9%

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

       3. times-frac 43.9%

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

       4. sqrt-prod 43.9%

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

       5. unpow2 43.9%

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

       6. sqrt-prod 30.7%

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

       7. add-sqr-sqrt 37.5%

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

       8. sqrt-prod 37.5%

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

       9. unpow2 37.5%

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

       10. sqrt-prod 32.0%

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

       11. add-sqr-sqrt 46.6%

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

   13. Applied egg-rr 46.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0 \cdot \sqrt{1 - \left(\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right) \cdot \sqrt{h}}{\ell}}}\right)}^{2}\\ \end{array} \]

Alternative 3: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M_m}{2 \cdot d_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0_m\\ \end{array} \end{array} \]

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 1 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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) 2e-10)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0)))))
    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 ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 2e-10) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D_m * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_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) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= 2d-10) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d_m * (m_m / (2.0d0 * d_m_1))) ** 2.0d0))))
    else
        tmp = w0_m
    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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 2e-10) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_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 (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 2e-10:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D_m * (M_m / (2.0 * d_m))), 2.0))))
	else:
		tmp = 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((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= 2e-10)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0)))));
	else
		tmp = w0_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) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= 2e-10)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D_m * (M_m / (2.0 * d_m))) ^ 2.0))));
	else
		tmp = w0_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[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 2e-10], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2.00000000000000007e-10

   1. Initial program 86.0%

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

   3. Simplified 85.6%

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

   if 2.00000000000000007e-10 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in M around 0 69.2%

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

4. Final simplification 84.5%

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

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;M_m \cdot D_m \leq 10^{-31}:\\ \;\;\;\;w0_m\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M_m}{2} \cdot \frac{D_m}{d_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 1 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) 1e-31)
    w0_m
    (*
     w0_m
     (sqrt (- 1.0 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m 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 tmp;
	if ((M_m * D_m) <= 1e-31) {
		tmp = w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow(((M_m / 2.0) * (D_m / 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) :: tmp
    if ((m_m * d_m) <= 1d-31) then
        tmp = w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * (((m_m / 2.0d0) * (d_m / 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 tmp;
	if ((M_m * D_m) <= 1e-31) {
		tmp = w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow(((M_m / 2.0) * (D_m / 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):
	tmp = 0
	if (M_m * D_m) <= 1e-31:
		tmp = w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow(((M_m / 2.0) * (D_m / 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)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-31)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / 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)
	tmp = 0.0;
	if ((M_m * D_m) <= 1e-31)
		tmp = w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M_m / 2.0) * (D_m / 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_] := N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-31], w0$95$m, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1e-31

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 75.3%

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

   if 1e-31 < (*.f64 M D)

   1. Initial program 77.6%

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

   3. Simplified 77.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \end{array} \]

Alternative 5: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \left(w0_m \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M_m}{d_m} \cdot \frac{D_m}{2}\right)}^{2}}{\ell}}\right) \end{array} \]

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 1 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 (sqrt (- 1.0 (/ (* h (pow (* (/ M_m d_m) (/ D_m 2.0)) 2.0)) 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) {
	return w0_s * (w0_m * sqrt((1.0 - ((h * pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l))));
}

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 * sqrt((1.0d0 - ((h * (((m_m / d_m_1) * (d_m / 2.0d0)) ** 2.0d0)) / l))))
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 * Math.sqrt((1.0 - ((h * Math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l))));
}

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 * math.sqrt((1.0 - ((h * math.pow(((M_m / d_m) * (D_m / 2.0)), 2.0)) / l))))

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 * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m / d_m) * Float64(D_m / 2.0)) ^ 2.0)) / l)))))
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 * sqrt((1.0 - ((h * (((M_m / d_m) * (D_m / 2.0)) ^ 2.0)) / l))));
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 * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.0%

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

   1. *-commutative 80.0%

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

   2. frac-times 80.0%

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

   3. associate-*l/ 80.4%

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

   4. associate-*l/ 86.0%

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

   5. associate-*l/ 85.9%

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

   6. *-commutative 85.9%

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

   7. times-frac 86.0%

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

5. Applied egg-rr 86.0%

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

6. Final simplification 86.0%

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

Alternative 6: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;M_m \cdot D_m \leq 2 \cdot 10^{+172}:\\ \;\;\;\;w0_m\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{{\left(M_m \cdot D_m\right)}^{2}}{\ell} \cdot \left(\frac{w0_m}{d_m} \cdot \frac{h}{d_m}\right)\right)\\ \end{array} \end{array} \]

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 1 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) 2e+172)
    w0_m
    (* -0.125 (* (/ (pow (* M_m D_m) 2.0) l) (* (/ w0_m d_m) (/ h 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) <= 2e+172) {
		tmp = w0_m;
	} else {
		tmp = -0.125 * ((pow((M_m * D_m), 2.0) / l) * ((w0_m / d_m) * (h / 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) <= 2d+172) then
        tmp = w0_m
    else
        tmp = (-0.125d0) * ((((m_m * d_m) ** 2.0d0) / l) * ((w0_m / d_m_1) * (h / 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) <= 2e+172) {
		tmp = w0_m;
	} else {
		tmp = -0.125 * ((Math.pow((M_m * D_m), 2.0) / l) * ((w0_m / d_m) * (h / 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) <= 2e+172:
		tmp = w0_m
	else:
		tmp = -0.125 * ((math.pow((M_m * D_m), 2.0) / l) * ((w0_m / d_m) * (h / 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) <= 2e+172)
		tmp = w0_m;
	else
		tmp = Float64(-0.125 * Float64(Float64((Float64(M_m * D_m) ^ 2.0) / l) * Float64(Float64(w0_m / d_m) * Float64(h / 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) <= 2e+172)
		tmp = w0_m;
	else
		tmp = -0.125 * ((((M_m * D_m) ^ 2.0) / l) * ((w0_m / d_m) * (h / 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], 2e+172], w0$95$m, N[(-0.125 * N[(N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(w0$95$m / d$95$m), $MachinePrecision] * N[(h / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.0000000000000002e172

   1. Initial program 80.5%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in M around 0 72.0%

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

   if 2.0000000000000002e172 < (*.f64 M D)

   1. Initial program 75.4%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in M around 0 60.2%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\sqrt{{D}^{2} \cdot \left({M}^{2} \cdot h\right)} \cdot \sqrt{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}}{{d}^{2} \cdot \ell}\right) \]

      2. pow2 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{{\left(\sqrt{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}\right)}^{2}}}{{d}^{2} \cdot \ell}\right) \]

      3. sqrt-prod 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\color{blue}{\left(\sqrt{{D}^{2}} \cdot \sqrt{{M}^{2} \cdot h}\right)}}^{2}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(\sqrt{\color{blue}{D \cdot D}} \cdot \sqrt{{M}^{2} \cdot h}\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      5. sqrt-prod 18.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(\color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)} \cdot \sqrt{{M}^{2} \cdot h}\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      6. add-sqr-sqrt 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(\color{blue}{D} \cdot \sqrt{{M}^{2} \cdot h}\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      7. sqrt-prod 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(D \cdot \color{blue}{\left(\sqrt{{M}^{2}} \cdot \sqrt{h}\right)}\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      8. unpow2 26.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(D \cdot \left(\sqrt{\color{blue}{M \cdot M}} \cdot \sqrt{h}\right)\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      9. sqrt-prod 22.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(D \cdot \left(\color{blue}{\left(\sqrt{M} \cdot \sqrt{M}\right)} \cdot \sqrt{h}\right)\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

      10. add-sqr-sqrt 29.9%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(D \cdot \left(\color{blue}{M} \cdot \sqrt{h}\right)\right)}^{2}}{{d}^{2} \cdot \ell}\right) \]

   6. Applied egg-rr 29.9%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{{\left(D \cdot \left(M \cdot \sqrt{h}\right)\right)}^{2}}}{{d}^{2} \cdot \ell}\right) \]

   7. Taylor expanded in D around inf 56.3%

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

      1. associate-*r* 56.2%

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

      2. *-commutative 56.2%

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

      3. times-frac 56.2%

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

      4. unpow2 56.2%

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

      5. unpow2 56.2%

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

      6. swap-sqr 56.2%

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

      7. unpow2 56.2%

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

   9. Simplified 56.2%

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

       1. *-commutative 56.2%

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

       2. unpow2 56.2%

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

       3. times-frac 56.8%

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

   11. Applied egg-rr 56.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+172}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{\ell} \cdot \left(\frac{w0}{d} \cdot \frac{h}{d}\right)\right)\\ \end{array} \]

Alternative 7: 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 1 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 80.0%

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

3. Simplified 80.0%

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

4. Taylor expanded in M around 0 67.2%

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

5. Final simplification 67.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023322 
(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))))))