Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 94.9%

Time: 24.3s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.6% 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: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(2 \cdot \log D_m + \left(2 \cdot \log M_m + 2 \cdot \log \left(\frac{1}{d_m}\right)\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    w0_s
    (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (log (* (/ h l) -0.25))
           (+
            (* 2.0 (log D_m))
            (+ (* 2.0 (log M_m)) (* 2.0 (log (/ 1.0 d_m)))))))))
       2.0)
      (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_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);
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * exp((0.25 * (log(((h / l) * -0.25)) + ((2.0 * log(D_m)) + ((2.0 * log(M_m)) + (2.0 * log((1.0 / d_m))))))))), 2.0);
	} else {
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	}
	return w0_s * tmp;
}

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);
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * (Math.log(((h / l) * -0.25)) + ((2.0 * Math.log(D_m)) + ((2.0 * Math.log(M_m)) + (2.0 * Math.log((1.0 / d_m))))))))), 2.0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_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)
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * (math.log(((h / l) * -0.25)) + ((2.0 * math.log(D_m)) + ((2.0 * math.log(M_m)) + (2.0 * math.log((1.0 / d_m))))))))), 2.0)
	else:
		tmp = w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_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(d_m / 0.5))
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(2.0 * log(D_m)) + Float64(Float64(2.0 * log(M_m)) + Float64(2.0 * log(Float64(1.0 / d_m))))))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_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);
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = (sqrt(w0_m) * exp((0.25 * (log(((h / l) * -0.25)) + ((2.0 * log(D_m)) + ((2.0 * log(M_m)) + (2.0 * log((1.0 / d_m))))))))) ^ 2.0;
	else
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, 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], (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[N[(1.0 / d$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $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)) < -inf.0

   1. Initial program 48.0%

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

   3. Simplified 48.0%

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

   6. Applied egg-rr 24.6%

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

   7. Taylor expanded in D around inf 12.7%

      \[\leadsto {\color{blue}{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{2} \]
   8. Step-by-step derivation

      1. exp-prod 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}}\right)}^{2} \]

      2. distribute-lft-neg-in 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      3. metadata-eval 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      4. times-frac 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      5. log-rec 12.7%

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

   9. Simplified 12.7%

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

   10. Taylor expanded in M around 0 5.6%

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

   11. Taylor expanded in d around inf 5.3%

       \[\leadsto {\left(\sqrt{w0} \cdot \color{blue}{e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h}{\ell}\right) + \left(2 \cdot \log D + \left(2 \cdot \log M + 2 \cdot \log \left(\frac{1}{d}\right)\right)\right)\right)}}\right)}^{2} \]

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

   1. Initial program 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. frac-times 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*l/ 97.4%

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

      5. div-inv 97.4%

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

      6. associate-*l* 97.4%

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

      7. associate-/r* 97.4%

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

      8. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

      3. associate-*r* 91.2%

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

      4. *-commutative 91.2%

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

      5. associate-*r* 91.2%

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

      6. clear-num 91.2%

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

      7. *-un-lft-identity 91.2%

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

      8. div-inv 92.3%

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

      9. unpow2 92.3%

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

      10. *-un-lft-identity 92.3%

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

      11. div-inv 92.2%

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

      12. times-frac 97.8%

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

      13. associate-*r/ 97.8%

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

      14. associate-*r/ 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in D around 0 91.9%

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

       1. associate-*r/ 91.9%

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

       2. associate-*r* 94.6%

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

       3. associate-*r* 94.6%

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

       4. *-commutative 94.6%

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

       5. associate-*r* 94.6%

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

       6. *-commutative 94.6%

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

       7. *-commutative 94.6%

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

       8. associate-*l/ 97.8%

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

       9. *-commutative 97.8%

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

       10. *-lft-identity 97.8%

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

       11. times-frac 97.8%

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

       12. /-rgt-identity 97.8%

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

   11. Simplified 97.8%

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

       1. expm1-log1p-u 97.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

       2. expm1-udef 97.5%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

       3. associate-*l/ 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

       4. associate-/l* 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

       5. associate-*r* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

       6. associate-/l* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

   13. Applied egg-rr 91.6%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 91.6%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 91.9%

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

       3. associate-/l* 93.0%

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

       4. associate-*r/ 92.9%

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

       5. associate-*l* 98.9%

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

       6. associate-*r/ 98.9%

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

   15. Simplified 98.9%

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

4. Final simplification 72.2%

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

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d_m + \left(2 \cdot \log D_m + 2 \cdot \log M_m\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    w0_s
    (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (log (* (/ h l) -0.25))
           (+ (* -2.0 (log d_m)) (+ (* 2.0 (log D_m)) (* 2.0 (log M_m))))))))
       2.0)
      (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_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);
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * exp((0.25 * (log(((h / l) * -0.25)) + ((-2.0 * log(d_m)) + ((2.0 * log(D_m)) + (2.0 * log(M_m)))))))), 2.0);
	} else {
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	}
	return w0_s * tmp;
}

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);
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * (Math.log(((h / l) * -0.25)) + ((-2.0 * Math.log(d_m)) + ((2.0 * Math.log(D_m)) + (2.0 * Math.log(M_m)))))))), 2.0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_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)
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * (math.log(((h / l) * -0.25)) + ((-2.0 * math.log(d_m)) + ((2.0 * math.log(D_m)) + (2.0 * math.log(M_m)))))))), 2.0)
	else:
		tmp = w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_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(d_m / 0.5))
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(-2.0 * log(d_m)) + Float64(Float64(2.0 * log(D_m)) + Float64(2.0 * log(M_m)))))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_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);
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = (sqrt(w0_m) * exp((0.25 * (log(((h / l) * -0.25)) + ((-2.0 * log(d_m)) + ((2.0 * log(D_m)) + (2.0 * log(M_m)))))))) ^ 2.0;
	else
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, 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], (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $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)) < -inf.0

   1. Initial program 48.0%

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

   3. Simplified 48.0%

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

   6. Applied egg-rr 24.6%

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

   7. Taylor expanded in D around inf 12.7%

      \[\leadsto {\color{blue}{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{2} \]
   8. Step-by-step derivation

      1. exp-prod 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}}\right)}^{2} \]

      2. distribute-lft-neg-in 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      3. metadata-eval 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      4. times-frac 12.7%

         \[\leadsto {\left(\sqrt{w0} \cdot {\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2} \]

      5. log-rec 12.7%

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

   9. Simplified 12.7%

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

   10. Taylor expanded in M around 0 5.6%

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

   11. Taylor expanded in d around 0 5.3%

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

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

   1. Initial program 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. frac-times 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*l/ 97.4%

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

      5. div-inv 97.4%

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

      6. associate-*l* 97.4%

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

      7. associate-/r* 97.4%

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

      8. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

      3. associate-*r* 91.2%

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

      4. *-commutative 91.2%

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

      5. associate-*r* 91.2%

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

      6. clear-num 91.2%

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

      7. *-un-lft-identity 91.2%

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

      8. div-inv 92.3%

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

      9. unpow2 92.3%

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

      10. *-un-lft-identity 92.3%

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

      11. div-inv 92.2%

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

      12. times-frac 97.8%

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

      13. associate-*r/ 97.8%

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

      14. associate-*r/ 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in D around 0 91.9%

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

       1. associate-*r/ 91.9%

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

       2. associate-*r* 94.6%

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

       3. associate-*r* 94.6%

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

       4. *-commutative 94.6%

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

       5. associate-*r* 94.6%

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

       6. *-commutative 94.6%

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

       7. *-commutative 94.6%

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

       8. associate-*l/ 97.8%

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

       9. *-commutative 97.8%

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

       10. *-lft-identity 97.8%

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

       11. times-frac 97.8%

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

       12. /-rgt-identity 97.8%

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

   11. Simplified 97.8%

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

       1. expm1-log1p-u 97.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

       2. expm1-udef 97.5%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

       3. associate-*l/ 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

       4. associate-/l* 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

       5. associate-*r* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

       6. associate-/l* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

   13. Applied egg-rr 91.6%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 91.6%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 91.9%

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

       3. associate-/l* 93.0%

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

       4. associate-*r/ 92.9%

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

       5. associate-*l* 98.9%

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

       6. associate-*r/ 98.9%

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

   15. Simplified 98.9%

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

4. Final simplification 72.2%

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

Alternative 3: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(-2 \cdot \log d_m + \log \left(-0.25 \cdot \frac{h \cdot {\left(M_m \cdot D_m\right)}^{2}}{\ell}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    w0_s
    (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (* -2.0 (log d_m))
           (log (* -0.25 (/ (* h (pow (* M_m D_m) 2.0)) l)))))))
       2.0)
      (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_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);
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + log((-0.25 * ((h * pow((M_m * D_m), 2.0)) / l))))))), 2.0);
	} else {
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	}
	return w0_s * tmp;
}

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);
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * ((-2.0 * Math.log(d_m)) + Math.log((-0.25 * ((h * Math.pow((M_m * D_m), 2.0)) / l))))))), 2.0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_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)
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * ((-2.0 * math.log(d_m)) + math.log((-0.25 * ((h * math.pow((M_m * D_m), 2.0)) / l))))))), 2.0)
	else:
		tmp = w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_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(d_m / 0.5))
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(Float64(-2.0 * log(d_m)) + log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / l))))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_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);
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = (sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + log((-0.25 * ((h * ((M_m * D_m) ^ 2.0)) / l))))))) ^ 2.0;
	else
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, 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], (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $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)) < -inf.0

   1. Initial program 48.0%

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

   3. Simplified 48.0%

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

   6. Applied egg-rr 24.6%

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

   7. Taylor expanded in d around 0 10.1%

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

      1. *-un-lft-identity 10.1%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \color{blue}{\left(1 \cdot \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)} + -2 \cdot \log d\right)}\right)}^{2} \]

      2. log-prod 10.1%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\color{blue}{\left(\log 1 + \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)} + -2 \cdot \log d\right)}\right)}^{2} \]

      3. metadata-eval 10.1%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(\color{blue}{0} + \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      4. distribute-lft-neg-in 10.1%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      5. metadata-eval 10.1%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      6. associate-*r* 11.3%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \left(-0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      7. unpow-prod-down 11.3%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \left(-0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      8. *-commutative 11.3%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell} \cdot -0.25\right)}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      9. *-commutative 11.3%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(0 + \log \left(\frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell} \cdot -0.25\right)\right) + -2 \cdot \log d\right)}\right)}^{2} \]

   9. Applied egg-rr 11.3%

      \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\color{blue}{\left(0 + \log \left(\frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell} \cdot -0.25\right)\right)} + -2 \cdot \log d\right)}\right)}^{2} \]

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

   1. Initial program 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. frac-times 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*l/ 97.4%

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

      5. div-inv 97.4%

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

      6. associate-*l* 97.4%

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

      7. associate-/r* 97.4%

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

      8. metadata-eval 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

      3. associate-*r* 91.2%

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

      4. *-commutative 91.2%

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

      5. associate-*r* 91.2%

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

      6. clear-num 91.2%

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

      7. *-un-lft-identity 91.2%

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

      8. div-inv 92.3%

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

      9. unpow2 92.3%

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

      10. *-un-lft-identity 92.3%

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

      11. div-inv 92.2%

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

      12. times-frac 97.8%

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

      13. associate-*r/ 97.8%

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

      14. associate-*r/ 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in D around 0 91.9%

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

       1. associate-*r/ 91.9%

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

       2. associate-*r* 94.6%

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

       3. associate-*r* 94.6%

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

       4. *-commutative 94.6%

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

       5. associate-*r* 94.6%

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

       6. *-commutative 94.6%

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

       7. *-commutative 94.6%

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

       8. associate-*l/ 97.8%

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

       9. *-commutative 97.8%

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

       10. *-lft-identity 97.8%

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

       11. times-frac 97.8%

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

       12. /-rgt-identity 97.8%

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

   11. Simplified 97.8%

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

       1. expm1-log1p-u 97.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

       2. expm1-udef 97.5%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

       3. associate-*l/ 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

       4. associate-/l* 97.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

       5. associate-*r* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

       6. associate-/l* 91.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

   13. Applied egg-rr 91.6%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 91.6%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 91.9%

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

       3. associate-/l* 93.0%

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

       4. associate-*r/ 92.9%

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

       5. associate-*l* 98.9%

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

       6. associate-*r/ 98.9%

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

   15. Simplified 98.9%

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

4. Final simplification 73.9%

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

Alternative 4: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;\frac{M_m \cdot D_m}{2 \cdot d_m} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M_m}^{2}}{\ell}\right) + \left(2 \cdot \log D_m + -2 \cdot \log d_m\right)\right)}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 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))))
   (*
    w0_s
    (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+298)
      (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_0))))))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (log (* -0.25 (/ (* h (pow M_m 2.0)) l)))
           (+ (* 2.0 (log D_m)) (* -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) / (d_m / 0.5);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+298) {
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	} else {
		tmp = pow((sqrt(w0_m) * exp((0.25 * (log((-0.25 * ((h * pow(M_m, 2.0)) / l))) + ((2.0 * log(D_m)) + (-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) :: tmp
    t_0 = (m_m * d_m) / (d_m_1 / 0.5d0)
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+298) then
        tmp = w0_m * sqrt((1.0d0 - (t_0 / (l / (h * t_0)))))
    else
        tmp = (sqrt(w0_m) * exp((0.25d0 * (log(((-0.25d0) * ((h * (m_m ** 2.0d0)) / l))) + ((2.0d0 * log(d_m)) + ((-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) / (d_m / 0.5);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+298) {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * (Math.log((-0.25 * ((h * Math.pow(M_m, 2.0)) / l))) + ((2.0 * Math.log(D_m)) + (-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) / (d_m / 0.5)
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+298:
		tmp = w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_0)))))
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * (math.log((-0.25 * ((h * math.pow(M_m, 2.0)) / l))) + ((2.0 * math.log(D_m)) + (-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(d_m / 0.5))
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+298)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_0))))));
	else
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(log(Float64(-0.25 * Float64(Float64(h * (M_m ^ 2.0)) / l))) + Float64(Float64(2.0 * log(D_m)) + 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) / (d_m / 0.5);
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+298)
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	else
		tmp = (sqrt(w0_m) * exp((0.25 * (log((-0.25 * ((h * (M_m ^ 2.0)) / l))) + ((2.0 * log(D_m)) + (-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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+298], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5.0000000000000003e298

   1. Initial program 80.4%

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

   3. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. frac-times 80.4%

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

      3. *-commutative 80.4%

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

      4. associate-*l/ 86.1%

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

      5. div-inv 86.1%

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

      6. associate-*l* 86.1%

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

      7. associate-/r* 86.1%

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

      8. metadata-eval 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-*r* 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-*r* 80.0%

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

      6. clear-num 80.0%

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

      7. *-un-lft-identity 80.0%

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

      8. div-inv 80.8%

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

      9. unpow2 80.8%

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

      10. *-un-lft-identity 80.8%

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

      11. div-inv 80.8%

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

      12. times-frac 86.9%

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

      13. associate-*r/ 86.9%

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

      14. associate-*r/ 86.9%

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

   8. Applied egg-rr 86.9%

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

   9. Taylor expanded in D around 0 80.6%

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

       1. associate-*r/ 80.6%

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

       2. associate-*r* 84.0%

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

       3. associate-*r* 84.0%

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

       4. *-commutative 84.0%

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

       5. associate-*r* 84.0%

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

       6. *-commutative 84.0%

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

       7. *-commutative 84.0%

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

       8. associate-*l/ 86.9%

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

       9. *-commutative 86.9%

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

       10. *-lft-identity 86.9%

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

       11. times-frac 86.9%

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

       12. /-rgt-identity 86.9%

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

   11. Simplified 86.9%

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

       1. expm1-log1p-u 86.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

       2. expm1-udef 86.5%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

       3. associate-*l/ 85.7%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

       4. associate-/l* 85.7%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

       5. associate-*r* 80.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

       6. associate-/l* 80.6%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

   13. Applied egg-rr 80.6%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 80.7%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 81.0%

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

       3. associate-/l* 82.1%

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

       4. associate-*r/ 82.1%

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

       5. associate-*l* 87.2%

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

       6. associate-*r/ 88.0%

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

   15. Simplified 88.0%

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

   if 5.0000000000000003e298 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 62.9%

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

   3. Simplified 67.1%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in d around 0 17.6%

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

      1. exp-prod 17.6%

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

      2. +-commutative 17.6%

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

      3. fma-def 17.6%

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

      4. distribute-lft-neg-in 17.6%

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

      5. metadata-eval 17.6%

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

      6. associate-*r* 17.6%

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

      7. *-commutative 17.6%

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

      8. unpow2 17.6%

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

      9. unpow2 17.6%

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

      10. swap-sqr 17.6%

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

      11. unpow2 17.6%

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

      12. *-commutative 17.6%

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

   9. Simplified 17.6%

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

   10. Taylor expanded in D around 0 8.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{\frac{d}{0.5}}}{\frac{\ell}{h \cdot \frac{M \cdot D}{\frac{d}{0.5}}}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;\frac{M_m \cdot D_m}{2 \cdot d_m} \leq 10^{+198}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(-2 \cdot \log d_m + \left(2 \cdot \log D_m + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M_m}^{2}\right)\right)\right)\right)}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 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))))
   (*
    w0_s
    (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1e+198)
      (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_0))))))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (* -2.0 (log d_m))
           (+ (* 2.0 (log D_m)) (log (* -0.25 (* (/ h l) (pow M_m 2.0)))))))))
       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);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+198) {
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	} else {
		tmp = pow((sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + ((2.0 * log(D_m)) + log((-0.25 * ((h / l) * pow(M_m, 2.0))))))))), 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) / (d_m_1 / 0.5d0)
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 1d+198) then
        tmp = w0_m * sqrt((1.0d0 - (t_0 / (l / (h * t_0)))))
    else
        tmp = (sqrt(w0_m) * exp((0.25d0 * (((-2.0d0) * log(d_m_1)) + ((2.0d0 * log(d_m)) + log(((-0.25d0) * ((h / l) * (m_m ** 2.0d0))))))))) ** 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);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+198) {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * ((-2.0 * Math.log(d_m)) + ((2.0 * Math.log(D_m)) + Math.log((-0.25 * ((h / l) * Math.pow(M_m, 2.0))))))))), 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)
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 1e+198:
		tmp = w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_0)))))
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * ((-2.0 * math.log(d_m)) + ((2.0 * math.log(D_m)) + math.log((-0.25 * ((h / l) * math.pow(M_m, 2.0))))))))), 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(d_m / 0.5))
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1e+198)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_0))))));
	else
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(Float64(-2.0 * log(d_m)) + Float64(Float64(2.0 * log(D_m)) + log(Float64(-0.25 * Float64(Float64(h / l) * (M_m ^ 2.0))))))))) ^ 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);
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+198)
		tmp = w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0)))));
	else
		tmp = (sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + ((2.0 * log(D_m)) + log((-0.25 * ((h / l) * (M_m ^ 2.0))))))))) ^ 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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e+198], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 1.00000000000000002e198

   1. Initial program 81.9%

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

   3. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. frac-times 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*l/ 87.8%

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

      5. div-inv 87.8%

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

      6. associate-*l* 87.7%

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

      7. associate-/r* 87.7%

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

      8. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

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

      1. associate-*l/ 81.8%

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

      2. *-commutative 81.8%

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

      3. associate-*r* 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-*r* 81.4%

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

      6. clear-num 81.4%

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

      7. *-un-lft-identity 81.4%

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

      8. div-inv 82.3%

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

      9. unpow2 82.3%

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

      10. *-un-lft-identity 82.3%

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

      11. div-inv 82.3%

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

      12. times-frac 88.1%

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

      13. associate-*r/ 88.1%

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

      14. associate-*r/ 88.2%

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

   8. Applied egg-rr 88.2%

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

   9. Taylor expanded in D around 0 81.5%

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

       1. associate-*r/ 81.5%

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

       2. associate-*r* 85.1%

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

       3. associate-*r* 85.1%

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

       4. *-commutative 85.1%

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

       5. associate-*r* 85.1%

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

       6. *-commutative 85.1%

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

       7. *-commutative 85.1%

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

       8. associate-*l/ 88.2%

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

       9. *-commutative 88.2%

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

       10. *-lft-identity 88.2%

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

       11. times-frac 88.2%

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

       12. /-rgt-identity 88.2%

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

   11. Simplified 88.2%

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

       1. expm1-log1p-u 87.7%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

       2. expm1-udef 87.7%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

       3. associate-*l/ 87.3%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

       4. associate-/l* 87.3%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

       5. associate-*r* 82.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

       6. associate-/l* 82.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

   13. Applied egg-rr 82.1%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 82.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 82.4%

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

       3. associate-/l* 83.1%

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

       4. associate-*r/ 83.1%

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

       5. associate-*l* 88.5%

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

       6. associate-*r/ 88.8%

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

   15. Simplified 88.8%

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

   if 1.00000000000000002e198 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 57.2%

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

   3. Simplified 60.3%

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

   6. Applied egg-rr 23.8%

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

   7. Taylor expanded in d around 0 13.1%

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

   8. Taylor expanded in D around 0 6.6%

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

      1. +-commutative 6.6%

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

      2. distribute-lft-neg-in 6.6%

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

      3. metadata-eval 6.6%

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

      4. associate-*r/ 6.6%

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

   10. Simplified 6.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+198}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{\frac{d}{0.5}}}{\frac{\ell}{h \cdot \frac{M \cdot D}{\frac{d}{0.5}}}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(-2 \cdot \log d + \left(2 \cdot \log D + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M}^{2}\right)\right)\right)\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0_s \cdot \left(w0_m \cdot \sqrt{1 - \frac{D_m \cdot \frac{M_m \cdot 0.5}{d_m}}{\ell} \cdot \left(0.5 \cdot \left(\frac{D_m}{d_m} \cdot \left(M_m \cdot h\right)\right)\right)}\right) \end{array} \]

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
     (*
      (/ (* D_m (/ (* M_m 0.5) d_m)) l)
      (* 0.5 (* (/ D_m d_m) (* M_m h)))))))))

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 - (((D_m * ((M_m * 0.5) / d_m)) / l) * (0.5 * ((D_m / d_m) * (M_m * h)))))));
}

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 - (((d_m * ((m_m * 0.5d0) / d_m_1)) / l) * (0.5d0 * ((d_m / d_m_1) * (m_m * h)))))))
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 - (((D_m * ((M_m * 0.5) / d_m)) / l) * (0.5 * ((D_m / d_m) * (M_m * h)))))));
}

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 - (((D_m * ((M_m * 0.5) / d_m)) / l) * (0.5 * ((D_m / d_m) * (M_m * h)))))))

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(Float64(D_m * Float64(Float64(M_m * 0.5) / d_m)) / l) * Float64(0.5 * Float64(Float64(D_m / d_m) * Float64(M_m * h))))))))
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 - (((D_m * ((M_m * 0.5) / d_m)) / l) * (0.5 * ((D_m / d_m) * (M_m * h)))))));
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[(N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 78.9%

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

   1. *-commutative 78.9%

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

   2. frac-times 78.9%

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

   3. *-commutative 78.9%

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

   4. associate-*l/ 84.1%

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

   5. div-inv 84.1%

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

   6. associate-*l* 84.4%

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

   7. associate-/r* 84.4%

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

   8. metadata-eval 84.4%

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

6. Applied egg-rr 84.4%

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

   1. associate-*l/ 79.2%

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

   2. *-commutative 79.2%

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

   3. associate-*r* 78.9%

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

   4. *-commutative 78.9%

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

   5. associate-*r* 78.8%

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

   6. clear-num 78.8%

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

   7. *-un-lft-identity 78.8%

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

   8. div-inv 79.6%

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

   9. unpow2 79.6%

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

   10. *-un-lft-identity 79.6%

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

   11. div-inv 79.6%

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

   12. times-frac 85.1%

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

   13. associate-*r/ 85.1%

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

   14. associate-*r/ 85.2%

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

8. Applied egg-rr 85.2%

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

9. Taylor expanded in D around 0 79.3%

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

    1. associate-*r/ 79.3%

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

    2. associate-*r* 82.1%

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

    3. associate-*r* 82.1%

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

    4. *-commutative 82.1%

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

    5. associate-*r* 82.1%

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

    6. *-commutative 82.1%

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

    7. *-commutative 82.1%

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

    8. associate-*l/ 84.8%

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

    9. *-commutative 84.8%

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

    10. *-lft-identity 84.8%

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

    11. times-frac 85.2%

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

    12. /-rgt-identity 85.2%

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

11. Simplified 85.2%

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

12. Taylor expanded in h around 0 79.3%

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

    1. associate-/l* 80.2%

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

    2. associate-/r/ 80.5%

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

14. Simplified 80.5%

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

15. Final simplification 80.5%

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

Alternative 7: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D_m \cdot \frac{M_m \cdot 0.5}{d_m}\\ w0_s \cdot \left(w0_m \cdot \sqrt{1 - \frac{t_0}{\ell} \cdot \left(h \cdot t_0\right)}\right) \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (/ (* M_m 0.5) d_m))))
   (* w0_s (* w0_m (sqrt (- 1.0 (* (/ t_0 l) (* h t_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 = D_m * ((M_m * 0.5) / d_m);
	return w0_s * (w0_m * sqrt((1.0 - ((t_0 / l) * (h * t_0)))));
}

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
    t_0 = d_m * ((m_m * 0.5d0) / d_m_1)
    code = w0_s * (w0_m * sqrt((1.0d0 - ((t_0 / l) * (h * t_0)))))
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 = D_m * ((M_m * 0.5) / d_m);
	return w0_s * (w0_m * Math.sqrt((1.0 - ((t_0 / l) * (h * t_0)))));
}

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 = D_m * ((M_m * 0.5) / d_m)
	return w0_s * (w0_m * math.sqrt((1.0 - ((t_0 / l) * (h * t_0)))))

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(D_m * Float64(Float64(M_m * 0.5) / d_m))
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(h * t_0))))))
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)
	t_0 = D_m * ((M_m * 0.5) / d_m);
	tmp = w0_s * (w0_m * sqrt((1.0 - ((t_0 / l) * (h * t_0)))));
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[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(t$95$0 / l), $MachinePrecision] * N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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 78.9%

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

   1. *-commutative 78.9%

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

   2. frac-times 78.9%

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

   3. *-commutative 78.9%

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

   4. associate-*l/ 84.1%

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

   5. div-inv 84.1%

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

   6. associate-*l* 84.4%

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

   7. associate-/r* 84.4%

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

   8. metadata-eval 84.4%

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

6. Applied egg-rr 84.4%

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

   1. associate-*l/ 79.2%

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

   2. *-commutative 79.2%

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

   3. associate-*r* 78.9%

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

   4. *-commutative 78.9%

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

   5. associate-*r* 78.8%

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

   6. clear-num 78.8%

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

   7. *-un-lft-identity 78.8%

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

   8. div-inv 79.6%

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

   9. unpow2 79.6%

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

   10. *-un-lft-identity 79.6%

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

   11. div-inv 79.6%

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

   12. times-frac 85.1%

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

   13. associate-*r/ 85.1%

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

   14. associate-*r/ 85.2%

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

8. Applied egg-rr 85.2%

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

9. Taylor expanded in D around 0 79.3%

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

    1. associate-*r/ 79.3%

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

    2. associate-*r* 82.1%

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

    3. associate-*r* 82.1%

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

    4. *-commutative 82.1%

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

    5. associate-*r* 82.1%

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

    6. *-commutative 82.1%

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

    7. *-commutative 82.1%

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

    8. associate-*l/ 84.8%

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

    9. *-commutative 84.8%

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

    10. *-lft-identity 84.8%

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

    11. times-frac 85.2%

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

    12. /-rgt-identity 85.2%

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

11. Simplified 85.2%

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

12. Final simplification 85.2%

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

Alternative 8: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{\frac{d_m}{0.5}}\\ w0_s \cdot \left(w0_m \cdot \sqrt{1 - \frac{t_0}{\frac{\ell}{h \cdot t_0}}}\right) \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* M_m D_m) (/ d_m 0.5))))
   (* w0_s (* w0_m (sqrt (- 1.0 (/ t_0 (/ l (* h t_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);
	return w0_s * (w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0))))));
}

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
    t_0 = (m_m * d_m) / (d_m_1 / 0.5d0)
    code = w0_s * (w0_m * sqrt((1.0d0 - (t_0 / (l / (h * t_0))))))
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);
	return w0_s * (w0_m * Math.sqrt((1.0 - (t_0 / (l / (h * t_0))))));
}

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)
	return w0_s * (w0_m * math.sqrt((1.0 - (t_0 / (l / (h * t_0))))))

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(d_m / 0.5))
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 / Float64(l / Float64(h * t_0)))))))
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)
	t_0 = (M_m * D_m) / (d_m / 0.5);
	tmp = w0_s * (w0_m * sqrt((1.0 - (t_0 / (l / (h * t_0))))));
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[(d$95$m / 0.5), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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 78.9%

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

   1. *-commutative 78.9%

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

   2. frac-times 78.9%

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

   3. *-commutative 78.9%

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

   4. associate-*l/ 84.1%

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

   5. div-inv 84.1%

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

   6. associate-*l* 84.4%

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

   7. associate-/r* 84.4%

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

   8. metadata-eval 84.4%

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

6. Applied egg-rr 84.4%

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

   1. associate-*l/ 79.2%

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

   2. *-commutative 79.2%

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

   3. associate-*r* 78.9%

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

   4. *-commutative 78.9%

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

   5. associate-*r* 78.8%

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

   6. clear-num 78.8%

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

   7. *-un-lft-identity 78.8%

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

   8. div-inv 79.6%

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

   9. unpow2 79.6%

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

   10. *-un-lft-identity 79.6%

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

   11. div-inv 79.6%

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

   12. times-frac 85.1%

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

   13. associate-*r/ 85.1%

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

   14. associate-*r/ 85.2%

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

8. Applied egg-rr 85.2%

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

9. Taylor expanded in D around 0 79.3%

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

    1. associate-*r/ 79.3%

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

    2. associate-*r* 82.1%

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

    3. associate-*r* 82.1%

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

    4. *-commutative 82.1%

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

    5. associate-*r* 82.1%

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

    6. *-commutative 82.1%

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

    7. *-commutative 82.1%

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

    8. associate-*l/ 84.8%

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

    9. *-commutative 84.8%

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

    10. *-lft-identity 84.8%

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

    11. times-frac 85.2%

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

    12. /-rgt-identity 85.2%

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

11. Simplified 85.2%

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

    1. expm1-log1p-u 84.7%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)\right)} \]

    2. expm1-udef 84.7%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{D \cdot \frac{M \cdot 0.5}{d}}{\ell} \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}\right)} - 1\right)} \]

    3. associate-*l/ 84.0%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}}\right)} - 1\right) \]

    4. associate-/l* 84.0%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right) \cdot \left(h \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)}{\ell}}\right)} - 1\right) \]

    5. associate-*r* 79.4%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \color{blue}{\left(\left(h \cdot D\right) \cdot \frac{M \cdot 0.5}{d}\right)}}{\ell}}\right)} - 1\right) \]

    6. associate-/l* 79.4%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}{\ell}}\right)} - 1\right) \]

13. Applied egg-rr 79.4%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)} - 1\right)} \]
14. Step-by-step derivation

    1. expm1-def 79.4%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(\left(h \cdot D\right) \cdot \frac{M}{\frac{d}{0.5}}\right)}{\ell}}\right)\right)} \]

    2. expm1-log1p 79.8%

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

    3. associate-/l* 80.7%

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

    4. associate-*r/ 80.3%

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

    5. associate-*l* 85.0%

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

    6. associate-*r/ 85.7%

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

15. Simplified 85.7%

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

16. Final simplification 85.7%

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{\frac{d}{0.5}}}{\frac{\ell}{h \cdot \frac{M \cdot D}{\frac{d}{0.5}}}}} \]
17. Add Preprocessing

Alternative 9: 68.0% 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 78.9%

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

3. Simplified 78.9%

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

5. Taylor expanded in D around 0 64.3%

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

6. Final simplification 64.3%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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))))))