Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.8% → 87.7%

Time: 16.9s

Alternatives: 8

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: 81.8% 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: 87.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \leq 5 \cdot 10^{+220}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{D}{\frac{d_m}{M}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d_m, \log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d_m)) 2.0) 5e+220)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ D (/ d_m M))) 2.0) h) l))))
   (pow
    (*
     (cbrt w0)
     (pow
      (exp 0.16666666666666666)
      (fma -2.0 (log d_m) (log (* -0.25 (/ (* h (pow (* M D) 2.0)) l))))))
    3.0)))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (pow(((M * D) / (2.0 * d_m)), 2.0) <= 5e+220) {
		tmp = w0 * sqrt((1.0 - ((pow((0.5 * (D / (d_m / M))), 2.0) * h) / l)));
	} else {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), fma(-2.0, log(d_m), log((-0.25 * ((h * pow((M * D), 2.0)) / l)))))), 3.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) <= 5e+220)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(D / Float64(d_m / M))) ^ 2.0) * h) / l))));
	else
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ fma(-2.0, log(d_m), log(Float64(-0.25 * Float64(Float64(h * (Float64(M * D) ^ 2.0)) / l)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+220], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(D / N[(d$95$m / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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 * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0000000000000002e220

   1. Initial program 91.5%

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

   3. Simplified 90.4%

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

      1. associate-*r/ 96.4%

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

      2. associate-*l/ 96.6%

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

      3. div-inv 96.6%

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

      4. associate-*l* 96.4%

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

      5. associate-/r* 96.4%

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

      6. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

   6. Taylor expanded in M around 0 96.6%

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

      1. associate-/l* 96.9%

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

   8. Simplified 96.9%

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

   if 5.0000000000000002e220 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 50.3%

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

   3. Simplified 49.0%

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

      1. associate-*r/ 53.8%

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

      2. associate-*l/ 55.2%

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

      3. div-inv 55.2%

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

      4. associate-*l* 55.2%

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

      5. associate-/r* 55.2%

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

      6. metadata-eval 55.2%

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

   5. Applied egg-rr 55.2%

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

      1. add-sqr-sqrt 24.4%

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

      2. pow2 24.4%

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

      3. sqrt-prod 24.4%

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

      4. unpow2 24.4%

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

      5. sqrt-prod 13.7%

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

      6. add-sqr-sqrt 27.2%

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

      7. associate-*r/ 27.2%

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

   7. Applied egg-rr 27.2%

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

      1. add-cube-cbrt 27.0%

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

      2. pow3 27.0%

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

   9. Applied egg-rr 55.2%

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

   10. Taylor expanded in d around 0 15.5%

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

       1. unpow1/3 31.3%

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

       2. *-lft-identity 31.3%

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

       3. exp-prod 31.0%

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

       4. +-commutative 31.0%

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

       5. fma-def 31.0%

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

       6. distribute-lft-neg-in 31.0%

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

       7. metadata-eval 31.0%

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

       8. associate-*r* 29.8%

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

       9. *-commutative 29.8%

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

       10. unpow2 29.8%

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

       11. unpow2 29.8%

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

       12. swap-sqr 38.3%

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

       13. unpow2 38.3%

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

       14. *-commutative 38.3%

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

   12. Simplified 38.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+220}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 2: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ w0 \cdot \sqrt{1 - h \cdot \frac{0.25}{\frac{\ell}{{\left(\frac{D}{\frac{d_m}{M}}\right)}^{2}}}} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ 0.25 (/ l (pow (/ D (/ d_m M)) 2.0))))))))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (h * (0.25 / (l / pow((D / (d_m / M)), 2.0))))));
}

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0 * sqrt((1.0d0 - (h * (0.25d0 / (l / ((d / (d_m / m)) ** 2.0d0))))))
end function

Java

d_m = Math.abs(d);
assert w0 < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (h * (0.25 / (l / Math.pow((D / (d_m / M)), 2.0))))));
}

Python

d_m = math.fabs(d)
[w0, M, D, h, l, d_m] = sort([w0, M, D, h, l, d_m])
def code(w0, M, D, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h * (0.25 / (l / math.pow((D / (d_m / M)), 2.0))))))

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(0.25 / Float64(l / (Float64(D / Float64(d_m / M)) ^ 2.0)))))))
end

MATLAB

d_m = abs(d);
w0, M, D, h, l, d_m = num2cell(sort([w0, M, D, h, l, d_m])){:}
function tmp = code(w0, M, D, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h * (0.25 / (l / ((D / (d_m / M)) ^ 2.0))))));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(0.25 / N[(l / N[Power[N[(D / N[(d$95$m / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 79.1%

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

   1. expm1-log1p-u 61.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}} \]

   2. expm1-udef 61.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{2 \cdot d} \cdot D\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}} \]

   3. log1p-udef 61.3%

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

   4. add-exp-log 79.1%

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

   5. +-commutative 79.1%

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

   6. associate-*l/ 80.2%

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

   7. div-inv 80.2%

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

   8. associate-*l* 79.4%

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

   9. associate-/r* 79.4%

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

   10. metadata-eval 79.4%

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

5. Applied egg-rr 79.4%

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

   1. associate--l+ 79.4%

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

   2. metadata-eval 79.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 79.4%

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

   4. associate-*r/ 85.1%

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

   5. associate-*l/ 84.7%

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

   6. *-commutative 84.7%

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

   7. associate-*r* 84.9%

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

   8. *-commutative 84.9%

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

   9. associate-*l/ 84.9%

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

   10. *-commutative 84.9%

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

   11. associate-*r/ 84.9%

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

   12. associate-*r/ 84.4%

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

7. Simplified 84.4%

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

   1. expm1-log1p-u 84.1%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}}\right)\right)} \]

   2. expm1-udef 84.0%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}}\right)} - 1\right)} \]

   3. unpow-prod-down 84.0%

      \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{\color{blue}{{0.5}^{2} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell}}\right)} - 1\right) \]

   4. metadata-eval 84.0%

      \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{\color{blue}{0.25} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}\right)} - 1\right) \]

   5. associate-*r/ 84.7%

      \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{0.25 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}}\right)} - 1\right) \]

9. Applied egg-rr 84.7%

   \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 84.7%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - h \cdot \frac{0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right)\right)} \]

    2. expm1-log1p 84.9%

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

    3. associate-/l* 84.9%

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

    4. associate-/l* 84.7%

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

11. Simplified 84.7%

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

12. Final simplification 84.7%

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

Alternative 3: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0 * sqrt((1.0d0 - ((((0.5d0 * (d / (d_m / m))) ** 2.0d0) * h) / l)))
end function

Java

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

Python

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

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(D / Float64(d_m / M))) ^ 2.0) * h) / l))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 79.1%

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

   1. associate-*r/ 84.7%

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

   2. associate-*l/ 85.3%

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

   3. div-inv 85.3%

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

   4. associate-*l* 85.1%

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

   5. associate-/r* 85.1%

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

   6. metadata-eval 85.1%

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

5. Applied egg-rr 85.1%

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

6. Taylor expanded in M around 0 85.3%

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

   1. associate-/l* 85.1%

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

8. Simplified 85.1%

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

9. Final simplification 85.1%

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

Alternative 4: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-311}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(M \cdot \frac{D}{d_m}\right)}^{2} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (/ h l) -1e-311)
   (+ w0 (* -0.125 (* (pow (* M (/ D d_m)) 2.0) (* w0 (/ h l)))))
   w0))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-311) {
		tmp = w0 + (-0.125 * (pow((M * (D / d_m)), 2.0) * (w0 * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if ((h / l) <= (-1d-311)) then
        tmp = w0 + ((-0.125d0) * (((m * (d / d_m)) ** 2.0d0) * (w0 * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
assert w0 < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-311) {
		tmp = w0 + (-0.125 * (Math.pow((M * (D / d_m)), 2.0) * (w0 * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
[w0, M, D, h, l, d_m] = sort([w0, M, D, h, l, d_m])
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (h / l) <= -1e-311:
		tmp = w0 + (-0.125 * (math.pow((M * (D / d_m)), 2.0) * (w0 * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1e-311)
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(M * Float64(D / d_m)) ^ 2.0) * Float64(w0 * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
w0, M, D, h, l, d_m = num2cell(sort([w0, M, D, h, l, d_m])){:}
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1e-311)
		tmp = w0 + (-0.125 * (((M * (D / d_m)) ^ 2.0) * (w0 * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-311], N[(w0 + N[(-0.125 * N[(N[Power[N[(M * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(w0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.99999999999948e-312

   1. Initial program 76.0%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in M around 0 53.2%

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

      1. add-sqr-sqrt 40.6%

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

      2. pow2 40.6%

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

      3. sqrt-prod 37.2%

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

      4. unpow2 37.2%

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

      5. sqrt-prod 16.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. sqrt-prod 28.6%

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

      8. unpow2 28.6%

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

      9. sqrt-prod 18.4%

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

      10. add-sqr-sqrt 31.4%

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

   6. Applied egg-rr 31.4%

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

   7. Taylor expanded in D around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. times-frac 51.9%

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

      3. unpow2 51.9%

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

      4. unpow2 51.9%

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

      5. swap-sqr 58.9%

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

      6. unpow2 58.9%

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

      7. times-frac 66.7%

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

      8. associate-*l/ 66.7%

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

      9. associate-*l/ 66.7%

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

      10. unpow2 66.7%

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

      11. *-commutative 66.7%

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

   9. Simplified 66.7%

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

       1. expm1-log1p-u 55.2%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\right)} \]

       2. expm1-udef 54.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right)} \]

       3. associate-/l* 53.4%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right)} - 1\right) \]

   11. Applied egg-rr 53.4%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 53.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\right)} \]

       2. expm1-log1p 65.2%

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

       3. associate-/r/ 67.1%

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

   13. Simplified 67.1%

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

   if -9.99999999999948e-312 < (/.f64 h l)

   1. Initial program 86.0%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in M around 0 91.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-311}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d_m}\right)}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= (/ h l) -5e-324)
   (+ w0 (* -0.125 (/ (* (pow (* M (/ D d_m)) 2.0) (* w0 h)) l)))
   w0))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 + (-0.125 * ((pow((M * (D / d_m)), 2.0) * (w0 * h)) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if ((h / l) <= (-5d-324)) then
        tmp = w0 + ((-0.125d0) * ((((m * (d / d_m)) ** 2.0d0) * (w0 * h)) / l))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
assert w0 < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-324) {
		tmp = w0 + (-0.125 * ((Math.pow((M * (D / d_m)), 2.0) * (w0 * h)) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
[w0, M, D, h, l, d_m] = sort([w0, M, D, h, l, d_m])
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (h / l) <= -5e-324:
		tmp = w0 + (-0.125 * ((math.pow((M * (D / d_m)), 2.0) * (w0 * h)) / l))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -5e-324)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64((Float64(M * Float64(D / d_m)) ^ 2.0) * Float64(w0 * h)) / l)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
w0, M, D, h, l, d_m = num2cell(sort([w0, M, D, h, l, d_m])){:}
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -5e-324)
		tmp = w0 + (-0.125 * ((((M * (D / d_m)) ^ 2.0) * (w0 * h)) / l));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-324], N[(w0 + N[(-0.125 * N[(N[(N[Power[N[(M * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.94066e-324

   1. Initial program 75.7%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in M around 0 52.5%

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

      1. add-sqr-sqrt 40.1%

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

      2. pow2 40.1%

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

      3. sqrt-prod 36.8%

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

      4. unpow2 36.8%

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

      5. sqrt-prod 16.2%

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

      6. add-sqr-sqrt 37.6%

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

      7. sqrt-prod 28.9%

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

      8. unpow2 28.9%

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

      9. sqrt-prod 18.1%

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

      10. add-sqr-sqrt 31.7%

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

   6. Applied egg-rr 31.7%

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

   7. Taylor expanded in D around 0 52.5%

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

      1. associate-*r* 52.5%

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

      2. times-frac 51.2%

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

      3. unpow2 51.2%

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

      4. unpow2 51.2%

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

      5. swap-sqr 58.8%

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

      6. unpow2 58.8%

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

      7. times-frac 66.5%

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

      8. associate-*l/ 66.4%

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

      9. associate-*l/ 66.4%

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

      10. unpow2 66.4%

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

      11. *-commutative 66.4%

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

   9. Simplified 66.4%

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

       1. associate-*r/ 69.6%

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

   11. Applied egg-rr 69.6%

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

   if -4.94066e-324 < (/.f64 h l)

   1. Initial program 86.7%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in M around 0 92.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \left(w0 \cdot h\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 73.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.6 \cdot 10^{+122}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d_m}\right)}^{2} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= D 2.6e+122)
   w0
   (+ w0 (* -0.125 (* (pow (/ (* M D) d_m) 2.0) (/ (* w0 h) l))))))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 2.6e+122) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow(((M * D) / d_m), 2.0) * ((w0 * h) / l)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (d <= 2.6d+122) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m * d) / d_m) ** 2.0d0) * ((w0 * h) / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
assert w0 < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 2.6e+122) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow(((M * D) / d_m), 2.0) * ((w0 * h) / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
[w0, M, D, h, l, d_m] = sort([w0, M, D, h, l, d_m])
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if D <= 2.6e+122:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow(((M * D) / d_m), 2.0) * ((w0 * h) / l)))
	return tmp

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (D <= 2.6e+122)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(Float64(M * D) / d_m) ^ 2.0) * Float64(Float64(w0 * h) / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
w0, M, D, h, l, d_m = num2cell(sort([w0, M, D, h, l, d_m])){:}
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (D <= 2.6e+122)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M * D) / d_m) ^ 2.0) * ((w0 * h) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[D, 2.6e+122], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.60000000000000007e122

   1. Initial program 81.5%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in M around 0 70.6%

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

   if 2.60000000000000007e122 < D

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in M around 0 32.0%

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

      1. add-sqr-sqrt 14.0%

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

      2. pow2 14.0%

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

      3. sqrt-prod 14.0%

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

      4. unpow2 14.0%

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

      5. sqrt-prod 31.6%

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

      6. add-sqr-sqrt 31.6%

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

      7. sqrt-prod 20.8%

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

      8. unpow2 20.8%

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

      9. sqrt-prod 13.8%

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

      10. add-sqr-sqrt 21.2%

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

   6. Applied egg-rr 21.2%

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

   7. Taylor expanded in D around 0 32.0%

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

      1. associate-*r* 32.0%

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

      2. times-frac 31.7%

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

      3. unpow2 31.7%

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

      4. unpow2 31.7%

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

      5. swap-sqr 49.5%

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

      6. unpow2 49.5%

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

      7. times-frac 70.3%

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

      8. associate-*l/ 66.8%

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

      9. associate-*l/ 66.8%

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

      10. unpow2 66.8%

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

      11. *-commutative 66.8%

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

   9. Simplified 66.8%

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

       1. associate-*r/ 70.3%

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

   11. Applied egg-rr 70.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.6 \cdot 10^{+122}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \]

Alternative 7: 70.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ [w0, M, D, h, l, d_m] = \mathsf{sort}([w0, M, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d_m}\\ \mathbf{if}\;M \leq 1.8 \cdot 10^{-94}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{w0 \cdot h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d_m)
 :precision binary64
 (let* ((t_0 (* M (/ D d_m))))
   (if (<= M 1.8e-94) w0 (+ w0 (* -0.125 (* (/ (* w0 h) l) (* t_0 t_0)))))))

C

d_m = fabs(d);
assert(w0 < M && M < D && D < h && h < l && l < d_m);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = M * (D / d_m);
	double tmp;
	if (M <= 1.8e-94) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) / l) * (t_0 * t_0)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / d_m)
    if (m <= 1.8d-94) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((w0 * h) / l) * (t_0 * t_0)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
assert w0 < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = M * (D / d_m);
	double tmp;
	if (M <= 1.8e-94) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) / l) * (t_0 * t_0)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
[w0, M, D, h, l, d_m] = sort([w0, M, D, h, l, d_m])
def code(w0, M, D, h, l, d_m):
	t_0 = M * (D / d_m)
	tmp = 0
	if M <= 1.8e-94:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((w0 * h) / l) * (t_0 * t_0)))
	return tmp

Julia

d_m = abs(d)
w0, M, D, h, l, d_m = sort([w0, M, D, h, l, d_m])
function code(w0, M, D, h, l, d_m)
	t_0 = Float64(M * Float64(D / d_m))
	tmp = 0.0
	if (M <= 1.8e-94)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(w0 * h) / l) * Float64(t_0 * t_0))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
w0, M, D, h, l, d_m = num2cell(sort([w0, M, D, h, l, d_m])){:}
function tmp_2 = code(w0, M, D, h, l, d_m)
	t_0 = M * (D / d_m);
	tmp = 0.0;
	if (M <= 1.8e-94)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((w0 * h) / l) * (t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 1.8e-94], w0, N[(w0 + N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8e-94

   1. Initial program 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in M around 0 69.9%

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

   if 1.8e-94 < M

   1. Initial program 76.1%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in M around 0 49.3%

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

      1. add-sqr-sqrt 34.9%

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

      2. pow2 34.9%

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

      3. sqrt-prod 32.3%

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

      4. unpow2 32.3%

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

      5. sqrt-prod 19.2%

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

      6. add-sqr-sqrt 33.5%

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

      7. sqrt-prod 33.5%

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

      8. unpow2 33.5%

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

      9. sqrt-prod 37.4%

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

      10. add-sqr-sqrt 37.4%

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

   6. Applied egg-rr 37.4%

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

   7. Taylor expanded in D around 0 49.3%

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

      1. associate-*r* 51.9%

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

      2. times-frac 48.0%

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

      3. unpow2 48.0%

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

      4. unpow2 48.0%

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

      5. swap-sqr 59.7%

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

      6. unpow2 59.7%

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

      7. times-frac 63.8%

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

      8. associate-*l/ 63.8%

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

      9. associate-*l/ 63.8%

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

      10. unpow2 63.8%

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

      11. *-commutative 63.8%

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

   9. Simplified 63.8%

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

       1. unpow2 63.8%

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

   11. Applied egg-rr 63.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-94}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{w0 \cdot h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\\ \end{array} \]

Alternative 8: 68.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 79.1%

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

4. Taylor expanded in M around 0 67.0%

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

5. Final simplification 67.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023326 
(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))))))