Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 92.7%

Time: 18.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(\frac{h \cdot -0.25}{\ell}\right) + 2 \cdot \log M_m\right) + \left(-2 \cdot \log d_m + 2 \cdot \log D_m\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D_m \cdot \left(M_m \cdot 0.5\right)}{d_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+253)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (+ (log (/ (* h -0.25) l)) (* 2.0 (log M_m)))
        (+ (* -2.0 (log d_m)) (* 2.0 (log D_m)))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0)) 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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((log(((h * -0.25) / l)) + (2.0 * log(M_m))) + ((-2.0 * log(d_m)) + (2.0 * log(D_m))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	return tmp;
}

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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((Math.log(((h * -0.25) / l)) + (2.0 * Math.log(M_m))) + ((-2.0 * Math.log(d_m)) + (2.0 * Math.log(D_m))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	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((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+253)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(log(Float64(Float64(h * -0.25) / l)) + Float64(2.0 * log(M_m))) + Float64(Float64(-2.0 * log(d_m)) + Float64(2.0 * log(D_m))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0)) / l))));
	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_] := 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], -1e+253], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(N[Log[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[M$95$m], $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], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -9.9999999999999994e252

   1. Initial program 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   3. Applied egg-rr 56.5%

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

   4. Taylor expanded in d around 0 12.2%

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

      1. unpow1/3 25.2%

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

      2. *-lft-identity 25.2%

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

      3. exp-prod 25.1%

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

      4. +-commutative 25.1%

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

      5. fma-def 25.1%

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

      6. distribute-lft-neg-in 25.1%

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

      7. metadata-eval 25.1%

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

      8. associate-*r* 26.5%

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

      9. unpow2 26.5%

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

      10. unpow2 26.5%

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

      11. swap-sqr 36.3%

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

      12. unpow2 36.3%

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

      13. associate-/l* 37.4%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in D around 0 13.0%

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

   8. Taylor expanded in M around 0 8.5%

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

      1. associate-*r/ 8.5%

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

   10. Simplified 8.5%

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

   if -9.9999999999999994e252 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 90.5%

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

      1. associate-*r/ 95.3%

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

      2. pow1 95.3%

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

      3. pow1 95.3%

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

      4. times-frac 94.7%

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

      5. div-inv 94.7%

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

      6. metadata-eval 94.7%

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

   3. Applied egg-rr 94.7%

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

      1. associate-*r/ 95.3%

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

   5. Applied egg-rr 95.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(\log \left(\frac{h \cdot -0.25}{\ell}\right) + 2 \cdot \log M\right) + \left(-2 \cdot \log d + 2 \cdot \log D\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d_m + \left(2 \cdot \log M_m + 2 \cdot \log D_m\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D_m \cdot \left(M_m \cdot 0.5\right)}{d_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+253)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (log (* (/ h l) -0.25))
        (+ (* -2.0 (log d_m)) (+ (* 2.0 (log M_m)) (* 2.0 (log D_m))))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0)) 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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log(((h / l) * -0.25)) + ((-2.0 * log(d_m)) + ((2.0 * log(M_m)) + (2.0 * log(D_m)))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	return tmp;
}

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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (Math.log(((h / l) * -0.25)) + ((-2.0 * Math.log(d_m)) + ((2.0 * Math.log(M_m)) + (2.0 * Math.log(D_m)))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	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((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+253)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * 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(D_m)))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0)) / l))));
	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_] := 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], -1e+253], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * 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[D$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -9.9999999999999994e252

   1. Initial program 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   3. Applied egg-rr 56.5%

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

   4. Taylor expanded in d around 0 12.2%

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

      1. unpow1/3 25.2%

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

      2. *-lft-identity 25.2%

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

      3. exp-prod 25.1%

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

      4. +-commutative 25.1%

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

      5. fma-def 25.1%

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

      6. distribute-lft-neg-in 25.1%

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

      7. metadata-eval 25.1%

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

      8. associate-*r* 26.5%

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

      9. unpow2 26.5%

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

      10. unpow2 26.5%

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

      11. swap-sqr 36.3%

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

      12. unpow2 36.3%

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

      13. associate-/l* 37.4%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in D around 0 13.0%

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

   8. Taylor expanded in M around 0 8.5%

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

   if -9.9999999999999994e252 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 90.5%

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

      1. associate-*r/ 95.3%

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

      2. pow1 95.3%

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

      3. pow1 95.3%

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

      4. times-frac 94.7%

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

      5. div-inv 94.7%

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

      6. metadata-eval 94.7%

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

   3. Applied egg-rr 94.7%

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

      1. associate-*r/ 95.3%

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

   5. Applied egg-rr 95.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d + \left(2 \cdot \log M + 2 \cdot \log D\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \mathsf{fma}\left(-2, \log d_m, \log \left(\frac{-0.25 \cdot {\left(M_m \cdot D_m\right)}^{2}}{\frac{\ell}{h}}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D_m \cdot \left(M_m \cdot 0.5\right)}{d_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+253)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (fma
        -2.0
        (log d_m)
        (log (/ (* -0.25 (pow (* M_m D_m) 2.0)) (/ l h)))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0)) 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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * fma(-2.0, log(d_m), log(((-0.25 * pow((M_m * D_m), 2.0)) / (l / h))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	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((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+253)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * fma(-2.0, log(d_m), log(Float64(Float64(-0.25 * (Float64(M_m * D_m) ^ 2.0)) / Float64(l / h))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0)) / l))));
	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_] := 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], -1e+253], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(-2.0 * N[Log[d$95$m], $MachinePrecision] + N[Log[N[(N[(-0.25 * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -9.9999999999999994e252

   1. Initial program 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   3. Applied egg-rr 56.5%

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

   4. Taylor expanded in d around 0 12.2%

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

      1. unpow1/3 25.2%

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

      2. *-lft-identity 25.2%

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

      3. exp-prod 25.1%

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

      4. +-commutative 25.1%

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

      5. fma-def 25.1%

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

      6. distribute-lft-neg-in 25.1%

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

      7. metadata-eval 25.1%

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

      8. associate-*r* 26.5%

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

      9. unpow2 26.5%

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

      10. unpow2 26.5%

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

      11. swap-sqr 36.3%

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

      12. unpow2 36.3%

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

      13. associate-/l* 37.4%

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

   6. Simplified 37.4%

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

      1. add-exp-log 37.4%

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

      2. log-pow 37.4%

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

      3. associate-*r/ 37.4%

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

      4. *-commutative 37.4%

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

      5. rem-log-exp 37.8%

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

   8. Applied egg-rr 37.8%

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

   if -9.9999999999999994e252 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 90.5%

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

      1. associate-*r/ 95.3%

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

      2. pow1 95.3%

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

      3. pow1 95.3%

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

      4. times-frac 94.7%

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

      5. div-inv 94.7%

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

      6. metadata-eval 94.7%

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

   3. Applied egg-rr 94.7%

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

      1. associate-*r/ 95.3%

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

   5. Applied egg-rr 95.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot {\left(M \cdot D\right)}^{2}}{\frac{\ell}{h}}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{\mathsf{fma}\left(0.16666666666666666, \log \left(h \cdot \left(-0.25 \cdot \frac{{\left(M_m \cdot D_m\right)}^{2}}{\ell}\right)\right), \log d_m \cdot -0.3333333333333333\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D_m \cdot \left(M_m \cdot 0.5\right)}{d_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+253)
   (pow
    (*
     (cbrt w0)
     (exp
      (fma
       0.16666666666666666
       (log (* h (* -0.25 (/ (pow (* M_m D_m) 2.0) l))))
       (* (log d_m) -0.3333333333333333))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0)) 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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+253) {
		tmp = pow((cbrt(w0) * exp(fma(0.16666666666666666, log((h * (-0.25 * (pow((M_m * D_m), 2.0) / l)))), (log(d_m) * -0.3333333333333333)))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / l)));
	}
	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((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+253)
		tmp = Float64(cbrt(w0) * exp(fma(0.16666666666666666, log(Float64(h * Float64(-0.25 * Float64((Float64(M_m * D_m) ^ 2.0) / l)))), Float64(log(d_m) * -0.3333333333333333)))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0)) / l))));
	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_] := 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], -1e+253], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[Log[N[(h * N[(-0.25 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[d$95$m], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -9.9999999999999994e252

   1. Initial program 57.7%

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

      1. add-cube-cbrt 57.7%

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

      2. pow3 57.7%

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

   3. Applied egg-rr 56.5%

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

   4. Taylor expanded in d around 0 12.2%

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

      1. unpow1/3 25.2%

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

      2. *-lft-identity 25.2%

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

      3. exp-prod 25.1%

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

      4. +-commutative 25.1%

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

      5. fma-def 25.1%

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

      6. distribute-lft-neg-in 25.1%

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

      7. metadata-eval 25.1%

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

      8. associate-*r* 26.5%

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

      9. unpow2 26.5%

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

      10. unpow2 26.5%

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

      11. swap-sqr 36.3%

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

      12. unpow2 36.3%

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

      13. associate-/l* 37.4%

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

   6. Simplified 37.4%

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

   7. Taylor expanded in d around 0 25.2%

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

   9. Simplified 37.8%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{\mathsf{fma}\left(0.16666666666666666, \log \left(\left(-0.25 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\ell}\right) \cdot h\right), -0.3333333333333333 \cdot \log d\right)}}\right)}^{3} \]

   if -9.9999999999999994e252 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 90.5%

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

      1. associate-*r/ 95.3%

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

      2. pow1 95.3%

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

      3. pow1 95.3%

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

      4. times-frac 94.7%

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

      5. div-inv 94.7%

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

      6. metadata-eval 94.7%

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

   3. Applied egg-rr 94.7%

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

      1. associate-*r/ 95.3%

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

   5. Applied egg-rr 95.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{\mathsf{fma}\left(0.16666666666666666, \log \left(h \cdot \left(-0.25 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\ell}\right)\right), \log d \cdot -0.3333333333333333\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D_m}^{2} \cdot \left(h \cdot {M_m}^{2}\right)}{\ell \cdot {d_m}^{2}}}\\ \end{array} \end{array} \]

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 2e-29)
     (* w0 (sqrt (- 1.0 t_0)))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         0.25
         (/ (* (pow D_m 2.0) (* h (pow M_m 2.0))) (* l (pow d_m 2.0))))))))))

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 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 2e-29) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((pow(D_m, 2.0) * (h * pow(M_m, 2.0))) / (l * pow(d_m, 2.0))))));
	}
	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 = (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 2d-29) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d_m ** 2.0d0) * (h * (m_m ** 2.0d0))) / (l * (d_m_1 ** 2.0d0))))))
    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 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 2e-29) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((Math.pow(D_m, 2.0) * (h * Math.pow(M_m, 2.0))) / (l * Math.pow(d_m, 2.0))))));
	}
	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 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 2e-29:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((math.pow(D_m, 2.0) * (h * math.pow(M_m, 2.0))) / (l * math.pow(d_m, 2.0))))))
	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(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 2e-29)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64((D_m ^ 2.0) * Float64(h * (M_m ^ 2.0))) / Float64(l * (d_m ^ 2.0)))))));
	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 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 2e-29)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D_m ^ 2.0) * (h * (M_m ^ 2.0))) / (l * (d_m ^ 2.0))))));
	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[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, 2e-29], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $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)) < 1.99999999999999989e-29

   1. Initial program 86.5%

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

   if 1.99999999999999989e-29 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   2. Taylor expanded in M around 0 71.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{{D_m}^{2}}{\frac{{d_m}^{2}}{h \cdot {M_m}^{2}}}}{\ell}}\\ \end{array} \end{array} \]

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 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 2e-30)
     (* w0 (sqrt (- 1.0 t_0)))
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (* 0.25 (/ (pow D_m 2.0) (/ (pow d_m 2.0) (* h (pow M_m 2.0)))))
         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 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 2e-30) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * (pow(D_m, 2.0) / (pow(d_m, 2.0) / (h * pow(M_m, 2.0))))) / l)));
	}
	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 = (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 2d-30) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((d_m ** 2.0d0) / ((d_m_1 ** 2.0d0) / (h * (m_m ** 2.0d0))))) / l)))
    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 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 2e-30) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (Math.pow(D_m, 2.0) / (Math.pow(d_m, 2.0) / (h * Math.pow(M_m, 2.0))))) / l)));
	}
	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 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 2e-30:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (math.pow(D_m, 2.0) / (math.pow(d_m, 2.0) / (h * math.pow(M_m, 2.0))))) / l)))
	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 <= 2e-30)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64((D_m ^ 2.0) / Float64((d_m ^ 2.0) / Float64(h * (M_m ^ 2.0))))) / l))));
	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 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 2e-30)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * ((D_m ^ 2.0) / ((d_m ^ 2.0) / (h * (M_m ^ 2.0))))) / l)));
	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[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, 2e-30], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[Power[D$95$m, 2.0], $MachinePrecision] / N[(N[Power[d$95$m, 2.0], $MachinePrecision] / N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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)) < 2e-30

   1. Initial program 86.5%

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

   if 2e-30 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 5.6%

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

      1. associate-*r/ 53.4%

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

      2. pow1 53.4%

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

      3. pow1 53.4%

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

      4. times-frac 53.4%

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

      5. div-inv 53.4%

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

      6. metadata-eval 53.4%

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

   3. Applied egg-rr 53.4%

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

   4. Taylor expanded in M around 0 67.8%

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

      1. associate-/l* 73.4%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}}{\ell}} \]

   6. Simplified 73.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}}{\ell}}\\ \end{array} \]

Alternative 7: 86.9% accurate, 0.7× speedup

Math

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

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 (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 INFINITY) (* w0 (sqrt t_0)) 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) {
	double t_0 = 1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0;
	}
	return tmp;
}

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 = 1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0;
	}
	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 = 1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= math.inf:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0
	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(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = w0;
	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 = 1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0;
	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[(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]}, If[LessEqual[t$95$0, Infinity], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], w0]]

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))) < +inf.0

   1. Initial program 86.5%

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

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 0.0%

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

   2. Taylor expanded in M around 0 61.7%

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

4. Final simplification 84.9%

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

Alternative 8: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D_m \leq 3 \cdot 10^{-24}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M_m \cdot \left(D_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 (<= D_m 3e-24)
   w0
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0)))))))

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 tmp;
	if (D_m <= 3e-24) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M_m * (D_m * (0.5 / d_m))), 2.0))));
	}
	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) :: tmp
    if (d_m <= 3d-24) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0))))
    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 (D_m <= 3e-24) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M_m * (D_m * (0.5 / d_m))), 2.0))));
	}
	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):
	tmp = 0
	if D_m <= 3e-24:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M_m * (D_m * (0.5 / d_m))), 2.0))))
	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)
	tmp = 0.0
	if (D_m <= 3e-24)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) ^ 2.0)))));
	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)
	tmp = 0.0;
	if (D_m <= 3e-24)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * (D_m * (0.5 / d_m))) ^ 2.0))));
	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_] := If[LessEqual[D$95$m, 3e-24], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.99999999999999995e-24

   1. Initial program 83.4%

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

   2. Taylor expanded in M around 0 68.4%

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

   if 2.99999999999999995e-24 < D

   1. Initial program 70.4%

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

      1. div-inv 70.4%

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

      2. associate-*l* 66.7%

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

      3. associate-/r* 66.7%

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

      4. metadata-eval 66.7%

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

   3. Applied egg-rr 66.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3 \cdot 10^{-24}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \end{array} \]

Alternative 9: 86.4% accurate, 1.0× speedup

Math

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

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 (/ (* h (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0)) 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) {
	return w0 * sqrt((1.0 - ((h * pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / 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
    code = w0 * sqrt((1.0d0 - ((h * (((d_m * (m_m * 0.5d0)) / d_m_1) ** 2.0d0)) / 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) {
	return w0 * Math.sqrt((1.0 - ((h * Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / 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):
	return w0 * math.sqrt((1.0 - ((h * math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0)) / 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)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0)) / 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)
	tmp = w0 * sqrt((1.0 - ((h * (((D_m * (M_m * 0.5)) / d_m) ^ 2.0)) / 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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

   1. associate-*r/ 83.8%

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

   2. pow1 83.8%

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

   3. pow1 83.8%

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

   4. times-frac 83.0%

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

   5. div-inv 83.0%

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

   6. metadata-eval 83.0%

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

3. Applied egg-rr 83.0%

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

   1. associate-*r/ 83.8%

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

5. Applied egg-rr 83.8%

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

6. Final simplification 83.8%

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

Alternative 10: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= M_m 1.9e-26) w0 (log1p (expm1 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) {
	double tmp;
	if (M_m <= 1.9e-26) {
		tmp = w0;
	} else {
		tmp = log1p(expm1(w0));
	}
	return tmp;
}

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 <= 1.9e-26) {
		tmp = w0;
	} else {
		tmp = Math.log1p(Math.expm1(w0));
	}
	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 <= 1.9e-26:
		tmp = w0
	else:
		tmp = math.log1p(math.expm1(w0))
	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 <= 1.9e-26)
		tmp = w0;
	else
		tmp = log1p(expm1(w0));
	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_] := If[LessEqual[M$95$m, 1.9e-26], w0, N[Log[1 + N[(Exp[w0] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.90000000000000007e-26

   1. Initial program 82.9%

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

   2. Taylor expanded in M around 0 69.9%

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

   if 1.90000000000000007e-26 < M

   1. Initial program 74.9%

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

      1. add-cube-cbrt 73.9%

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

      2. pow3 74.0%

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

   3. Applied egg-rr 73.9%

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

   4. Taylor expanded in M around 0 22.5%

      \[\leadsto {\color{blue}{\left({w0}^{0.3333333333333333}\right)}}^{3} \]
   5. Step-by-step derivation

      1. unpow1/3 41.7%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{w0}\right)}}^{3} \]

   6. Simplified 41.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0}\right)}}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 42.6%

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

      2. log1p-expm1-u 46.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]

   8. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \]

Alternative 11: 68.9% accurate, 216.0× speedup

Math

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

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 80.8%

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

2. Taylor expanded in M around 0 62.6%

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

3. Final simplification 62.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023333 
(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))))))