Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.9%

Time: 19.1s

Alternatives: 7

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d_m, \log \left(-0.25 \cdot \frac{h \cdot {\left(M_m \cdot D_m\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D_m}{2} \cdot \frac{M_m}{d_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M_m}{\frac{\ell}{\frac{D_m \cdot 0.5}{d_m}}} \cdot \frac{M_m}{\frac{\frac{1}{h}}{D_m \cdot \frac{0.5}{d_m}}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (pow
      (*
       (cbrt w0)
       (pow
        (exp 0.16666666666666666)
        (fma
         -2.0
         (log d_m)
         (log (* -0.25 (/ (* h (pow (* M_m D_m) 2.0)) l))))))
      3.0)
     (if (<= t_0 2e-46)
       (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D_m 2.0) (/ M_m d_m)) 2.0)))))
       (*
        w0
        (sqrt
         (-
          1.0
          (*
           (/ M_m (/ l (/ (* D_m 0.5) d_m)))
           (/ M_m (/ (/ 1.0 h) (* D_m (/ 0.5 d_m))))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), fma(-2.0, log(d_m), log((-0.25 * ((h * pow((M_m * D_m), 2.0)) / l)))))), 3.0);
	} else if (t_0 <= 2e-46) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D_m / 2.0) * (M_m / d_m)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m / (l / ((D_m * 0.5) / d_m))) * (M_m / ((1.0 / h) / (D_m * (0.5 / d_m)))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ fma(-2.0, log(d_m), log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / l)))))) ^ 3.0;
	elseif (t_0 <= 2e-46)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m / 2.0) * Float64(M_m / d_m)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m / Float64(l / Float64(Float64(D_m * 0.5) / d_m))) * Float64(M_m / Float64(Float64(1.0 / h) / Float64(D_m * Float64(0.5 / d_m))))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(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]}, If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(-2.0 * N[Log[d$95$m], $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], 3.0], $MachinePrecision], If[LessEqual[t$95$0, 2e-46], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m / N[(l / N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(N[(1.0 / h), $MachinePrecision] / N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.1%

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

   3. Simplified 55.1%

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

      1. *-commutative 55.1%

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

      2. frac-times 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. add-sqr-sqrt 28.4%

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

      6. sqrt-unprod 28.4%

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

      7. *-commutative 28.4%

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

   6. Applied egg-rr 26.4%

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

      1. associate-*r* 26.5%

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

      2. *-commutative 26.5%

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

   8. Simplified 26.5%

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

      1. add-cube-cbrt 26.5%

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

      2. pow3 26.5%

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

   10. Applied egg-rr 55.0%

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

   11. Taylor expanded in d around 0 17.0%

       \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{3} \]
   12. Step-by-step derivation

       1. unpow1/3 31.0%

          \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       2. *-lft-identity 31.0%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       3. exp-prod 30.9%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}}\right)}^{3} \]

       4. +-commutative 30.9%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\color{blue}{\left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)}}\right)}^{3} \]

       5. fma-def 30.9%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\color{blue}{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)\right)}}\right)}^{3} \]

       6. distribute-lft-neg-in 30.9%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)}\right)\right)}\right)}^{3} \]

       7. metadata-eval 30.9%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)\right)}\right)}^{3} \]

       8. associate-*r* 32.7%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell}\right)\right)\right)}\right)}^{3} \]

       9. *-commutative 32.7%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot {D}^{2}\right)} \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       10. unpow2 32.7%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\left(\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}\right) \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       11. unpow2 32.7%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\left(\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       12. swap-sqr 35.5%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       13. unpow2 35.5%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\color{blue}{{\left(M \cdot D\right)}^{2}} \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       14. *-commutative 35.5%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot D\right)}^{2}}}{\ell}\right)\right)\right)}\right)}^{3} \]

       15. *-commutative 35.5%

           \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{h \cdot {\color{blue}{\left(D \cdot M\right)}}^{2}}{\ell}\right)\right)\right)}\right)}^{3} \]

   13. Simplified 35.5%

       \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right)\right)\right)}\right)}}^{3} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2.00000000000000005e-46

   1. Initial program 99.4%

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

   3. Simplified 97.6%

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

   if 2.00000000000000005e-46 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 9.9%

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

   3. Simplified 9.9%

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

      1. frac-times 9.9%

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

      2. *-commutative 9.9%

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

      3. clear-num 10.0%

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

      4. un-div-inv 20.0%

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

      5. div-inv 20.0%

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

      6. associate-*l* 20.0%

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

      7. associate-/r* 20.0%

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

      8. metadata-eval 20.0%

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

   6. Applied egg-rr 20.0%

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

      1. unpow2 20.0%

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

      2. div-inv 20.0%

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

      3. times-frac 77.5%

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

   8. Applied egg-rr 77.5%

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

      1. associate-/l* 77.4%

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

      2. associate-*r/ 77.4%

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

      3. associate-/r/ 77.4%

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

      4. /-rgt-identity 77.4%

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

      5. associate-*r/ 77.4%

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

   10. Simplified 77.4%

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

       1. associate-*r/ 77.4%

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

       2. pow1 77.4%

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

       3. metadata-eval 77.4%

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

       4. pow-flip 77.4%

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

       5. inv-pow 77.4%

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

       6. div-inv 77.4%

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

       7. associate-*r/ 77.4%

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

       8. associate-/l* 80.7%

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

       9. associate-*r/ 80.7%

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

   12. Applied egg-rr 80.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-46}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{\frac{\ell}{\frac{D \cdot 0.5}{d}}} \cdot \frac{M}{\frac{\frac{1}{h}}{D \cdot \frac{0.5}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))) <= 1.0d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d_m / 2.0d0) * (m_m / d_m_1)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m * ((d_m * 0.5d0) / d_m_1)) / l) * ((m_m * (d_m * (0.5d0 / d_m_1))) / (1.0d0 / h)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 1.0) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D_m / 2.0) * (M_m / d_m)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 1.0:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D_m / 2.0) * (M_m / d_m)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))))
	return tmp

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l))) <= 1.0)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D_m / 2.0) * (M_m / d_m)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   3. Simplified 97.5%

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

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

   1. Initial program 44.2%

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

   3. Simplified 44.2%

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

      1. frac-times 44.2%

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

      2. *-commutative 44.2%

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

      3. clear-num 44.2%

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

      4. un-div-inv 47.3%

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

      5. div-inv 47.3%

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

      6. associate-*l* 47.3%

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

      7. associate-/r* 47.3%

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

      8. metadata-eval 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. unpow2 47.3%

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

      2. div-inv 47.3%

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

      3. times-frac 71.2%

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

   8. Applied egg-rr 71.2%

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

      1. expm1-log1p-u 50.5%

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

      2. expm1-udef 43.1%

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

   10. Applied egg-rr 43.1%

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

       1. expm1-def 50.5%

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

       2. expm1-log1p 71.2%

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

       3. associate-*r/ 71.2%

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

   12. Simplified 71.2%

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

4. Final simplification 87.7%

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

Alternative 3: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 2.5 \cdot 10^{-172}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(M_m \cdot \frac{D_m \cdot 0.5}{d_m}\right)\right) \cdot \frac{M_m}{d_m \cdot \frac{\ell}{D_m \cdot 0.5}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 2.5e-172)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* h (* M_m (/ (* D_m 0.5) d_m)))
       (/ M_m (* d_m (/ l (* D_m 0.5))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.5e-172) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d_m))) * (M_m / (d_m * (l / (D_m * 0.5)))))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 2.5d-172) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (m_m * ((d_m * 0.5d0) / d_m_1))) * (m_m / (d_m_1 * (l / (d_m * 0.5d0)))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.5e-172) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d_m))) * (M_m / (d_m * (l / (D_m * 0.5)))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 2.5e-172:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d_m))) * (M_m / (d_m * (l / (D_m * 0.5)))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2.5e-172)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))) * Float64(M_m / Float64(d_m * Float64(l / Float64(D_m * 0.5))))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 2.5e-172)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d_m))) * (M_m / (d_m * (l / (D_m * 0.5)))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 2.5e-172], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(d$95$m * N[(l / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5e-172

   1. Initial program 83.8%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in D around 0 75.2%

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

   if 2.5e-172 < M

   1. Initial program 67.9%

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

   3. Simplified 65.5%

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

      1. frac-times 67.9%

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

      2. *-commutative 67.9%

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

      3. clear-num 67.9%

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

      4. un-div-inv 69.1%

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

      5. div-inv 69.1%

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

      6. associate-*l* 69.0%

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

      7. associate-/r* 69.0%

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

      8. metadata-eval 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. unpow2 69.0%

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

      2. div-inv 69.0%

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

      3. times-frac 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. associate-/l* 82.4%

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

      2. associate-*r/ 82.4%

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

      3. associate-/r/ 82.4%

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

      4. /-rgt-identity 82.4%

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

      5. associate-*r/ 82.5%

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

   10. Simplified 82.5%

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

       1. associate-/r/ 82.5%

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

   12. Applied egg-rr 82.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-172}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right) \cdot \frac{M}{d \cdot \frac{\ell}{D \cdot 0.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D_m \cdot 0.5}{d_m}\\ \mathbf{if}\;M_m \leq 3.3 \cdot 10^{-211}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M_m}{\frac{\ell}{t_0}} \cdot \left(h \cdot \left(M_m \cdot t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m 0.5) d_m)))
   (if (<= M_m 3.3e-211)
     w0
     (* w0 (sqrt (- 1.0 (* (/ M_m (/ l t_0)) (* h (* M_m t_0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m * 0.5) / d_m;
	double tmp;
	if (M_m <= 3.3e-211) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m / (l / t_0)) * (h * (M_m * t_0)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m * 0.5d0) / d_m_1
    if (m_m <= 3.3d-211) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m / (l / t_0)) * (h * (m_m * t_0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m * 0.5) / d_m;
	double tmp;
	if (M_m <= 3.3e-211) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m / (l / t_0)) * (h * (M_m * t_0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (D_m * 0.5) / d_m
	tmp = 0
	if M_m <= 3.3e-211:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m / (l / t_0)) * (h * (M_m * t_0)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m * 0.5) / d_m)
	tmp = 0.0
	if (M_m <= 3.3e-211)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m / Float64(l / t_0)) * Float64(h * Float64(M_m * t_0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (D_m * 0.5) / d_m;
	tmp = 0.0;
	if (M_m <= 3.3e-211)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m / (l / t_0)) * (h * (M_m * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]}, If[LessEqual[M$95$m, 3.3e-211], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3000000000000002e-211

   1. Initial program 83.1%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in D around 0 73.7%

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

   if 3.3000000000000002e-211 < M

   1. Initial program 71.3%

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

   3. Simplified 69.2%

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

      1. frac-times 71.3%

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

      2. *-commutative 71.3%

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

      3. clear-num 71.3%

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

      4. un-div-inv 72.3%

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

      5. div-inv 72.3%

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

      6. associate-*l* 72.3%

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

      7. associate-/r* 72.3%

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

      8. metadata-eval 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. unpow2 72.3%

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

      2. div-inv 72.3%

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

      3. times-frac 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l* 84.8%

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

      2. associate-*r/ 84.8%

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

      3. associate-/r/ 84.8%

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

      4. /-rgt-identity 84.8%

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

      5. associate-*r/ 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{-211}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{\frac{\ell}{\frac{D \cdot 0.5}{d}}} \cdot \left(h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (* D_m (/ 0.5 d_m)))))
   (* w0 (sqrt (- 1.0 (* (/ t_0 (/ 1.0 h)) (/ t_0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m * (0.5 / d_m));
	return w0 * sqrt((1.0 - ((t_0 / (1.0 / h)) * (t_0 / l))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = m_m * (d_m * (0.5d0 / d_m_1))
    code = w0 * sqrt((1.0d0 - ((t_0 / (1.0d0 / h)) * (t_0 / l))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m * (0.5 / d_m));
	return w0 * Math.sqrt((1.0 - ((t_0 / (1.0 / h)) * (t_0 / l))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = M_m * (D_m * (0.5 / d_m))
	return w0 * math.sqrt((1.0 - ((t_0 / (1.0 / h)) * (t_0 / l))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(M_m * Float64(D_m * Float64(0.5 / d_m)))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 / Float64(1.0 / h)) * Float64(t_0 / l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = M_m * (D_m * (0.5 / d_m));
	tmp = w0 * sqrt((1.0 - ((t_0 / (1.0 / h)) * (t_0 / l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.7%

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

3. Simplified 77.5%

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

   1. frac-times 78.7%

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

   2. *-commutative 78.7%

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

   3. clear-num 78.7%

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

   4. un-div-inv 79.9%

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

   5. div-inv 79.9%

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

   6. associate-*l* 79.0%

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

   7. associate-/r* 79.0%

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

   8. metadata-eval 79.0%

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

6. Applied egg-rr 79.0%

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

   1. unpow2 79.0%

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

   2. div-inv 79.0%

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

   3. times-frac 88.0%

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

8. Applied egg-rr 88.0%

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

9. Final simplification 88.0%

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

Alternative 6: 88.4% accurate, 1.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (*
     (/ (* M_m (/ (* D_m 0.5) d_m)) l)
     (/ (* M_m (* D_m (/ 0.5 d_m))) (/ 1.0 h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (((m_m * ((d_m * 0.5d0) / d_m_1)) / l) * ((m_m * (d_m * (0.5d0 / d_m_1))) / (1.0d0 / h)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(Float64(D_m * 0.5) / d_m)) / l) * Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) / Float64(1.0 / h))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (((M_m * ((D_m * 0.5) / d_m)) / l) * ((M_m * (D_m * (0.5 / d_m))) / (1.0 / h)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

3. Simplified 77.5%

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

   1. frac-times 78.7%

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

   2. *-commutative 78.7%

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

   3. clear-num 78.7%

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

   4. un-div-inv 79.9%

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

   5. div-inv 79.9%

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

   6. associate-*l* 79.0%

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

   7. associate-/r* 79.0%

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

   8. metadata-eval 79.0%

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

6. Applied egg-rr 79.0%

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

   1. unpow2 79.0%

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

   2. div-inv 79.0%

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

   3. times-frac 88.0%

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

8. Applied egg-rr 88.0%

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

   1. expm1-log1p-u 75.2%

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

   2. expm1-udef 71.8%

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

10. Applied egg-rr 71.8%

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

    1. expm1-def 75.2%

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

    2. expm1-log1p 88.0%

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

    3. associate-*r/ 88.1%

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

12. Simplified 88.1%

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

13. Final simplification 88.1%

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

Alternative 7: 68.4% accurate, 216.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 78.7%

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

3. Simplified 77.5%

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

5. Taylor expanded in D around 0 70.6%

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

6. Final simplification 70.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024016 
(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))))))