2-ancestry mixing, zero discriminant

Percentage Accurate: 75.6% → 98.7%

Time: 8.3s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Initial Program: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (* 0.5 g)) (cbrt a)))

C

double code(double g, double a) {
	return cbrt((0.5 * g)) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.5 * g)) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(0.5 * g)) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \sqrt[3]{g \cdot \frac{1}{2 \cdot a}} \]

   2. cbrt-prod N/A

      \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \]

   3. pow1/3 N/A

      \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{1}{2 \cdot a}\right)}^{\color{blue}{\frac{1}{3}}} \]

   4. *-commutative N/A

      \[\leadsto {\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{g}} \]

   5. inv-pow N/A

      \[\leadsto {\left({\left(2 \cdot a\right)}^{-1}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{g} \]

   6. pow-pow N/A

      \[\leadsto {\left(2 \cdot a\right)}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{\color{blue}{g}} \]

   7. unpow-prod-down N/A

      \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(-1 \cdot \frac{1}{3}\right)}\right) \cdot \sqrt[3]{\color{blue}{g}} \]

   8. metadata-eval N/A

      \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\frac{-1}{3}}\right) \cdot \sqrt[3]{g} \]

   9. metadata-eval N/A

      \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right) \cdot \sqrt[3]{g} \]

   10. pow-flip N/A

       \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{{a}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{g} \]

   11. pow1/3 N/A

       \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{\sqrt[3]{a}}\right) \cdot \sqrt[3]{g} \]

   12. div-inv N/A

       \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)}}{\sqrt[3]{a}} \cdot \sqrt[3]{\color{blue}{g}} \]

   13. associate-*l/ N/A

       \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

   16. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{3}\right)\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   18. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   19. cbrt-lowering-cbrt.f64 98.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. sqr-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)} \cdot {2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   2. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right)}^{2}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{-1}{3}}{2}\right)\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   5. metadata-eval 98.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{6}\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{{\left({2}^{-0.16666666666666666}\right)}^{2}} \cdot \sqrt[3]{g}}{\sqrt[3]{a}} \]
7. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto {\left({2}^{\frac{-1}{6}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \color{blue}{{\left({2}^{\frac{-1}{6}}\right)}^{2}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot {\color{blue}{\left({2}^{\frac{-1}{6}}\right)}}^{2} \]

   4. cbrt-div N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

   5. pow1/3 N/A

      \[\leadsto \frac{1}{{\left(\frac{a}{g}\right)}^{\frac{1}{3}}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

   6. pow-flip N/A

      \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot {\color{blue}{\left({2}^{\frac{-1}{6}}\right)}}^{2} \]

   7. metadata-eval N/A

      \[\leadsto {\left(\frac{a}{g}\right)}^{\frac{-1}{3}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

   8. metadata-eval N/A

      \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

   9. pow-pow N/A

      \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \cdot {2}^{\color{blue}{\left(\frac{-1}{6} \cdot 2\right)}} \]

   10. pow-prod-down N/A

       \[\leadsto {\left(\frac{a}{g} \cdot 2\right)}^{\color{blue}{\left(\frac{-1}{6} \cdot 2\right)}} \]

   11. metadata-eval N/A

       \[\leadsto {\left(\frac{a}{g} \cdot \frac{1}{\frac{1}{2}}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \]

   12. div-inv N/A

       \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\left(\color{blue}{\frac{-1}{6}} \cdot 2\right)} \]

   13. metadata-eval N/A

       \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\frac{-1}{3}} \]

   14. metadata-eval N/A

       \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{3}}\right)} \]

   15. pow-pow N/A

       \[\leadsto {\left({\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{-1}\right)}^{\color{blue}{\frac{1}{3}}} \]

   16. inv-pow N/A

       \[\leadsto {\left(\frac{1}{\frac{\frac{a}{g}}{\frac{1}{2}}}\right)}^{\frac{1}{3}} \]

   17. clear-num N/A

       \[\leadsto {\left(\frac{\frac{1}{2}}{\frac{a}{g}}\right)}^{\frac{1}{3}} \]

   18. pow1/3 N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{\frac{a}{g}}} \]

   19. associate-/r/ N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a} \cdot g} \]

   20. associate-*l/ N/A

       \[\leadsto \sqrt[3]{\frac{\frac{1}{2} \cdot g}{a}} \]

   21. cbrt-div N/A

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot g}}{\color{blue}{\sqrt[3]{a}}} \]

   22. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{2} \cdot g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

   23. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{2} \cdot g\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

   24. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   25. cbrt-lowering-cbrt.f64 98.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

8. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}}} \]
9. Add Preprocessing

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\frac{g}{\sqrt[3]{\frac{a \cdot \left(g \cdot g\right)}{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\frac{g}{2}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))))
   (if (<= t_0 -1e-318)
     (cbrt t_0)
     (if (<= t_0 2e-319)
       (/ g (cbrt (/ (* a (* g g)) 0.5)))
       (/ 1.0 (cbrt (/ a (/ g 2.0))))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = cbrt(t_0);
	} else if (t_0 <= 2e-319) {
		tmp = g / cbrt(((a * (g * g)) / 0.5));
	} else {
		tmp = 1.0 / cbrt((a / (g / 2.0)));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -1e-318) {
		tmp = Math.cbrt(t_0);
	} else if (t_0 <= 2e-319) {
		tmp = g / Math.cbrt(((a * (g * g)) / 0.5));
	} else {
		tmp = 1.0 / Math.cbrt((a / (g / 2.0)));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-318)
		tmp = cbrt(t_0);
	elseif (t_0 <= 2e-319)
		tmp = Float64(g / cbrt(Float64(Float64(a * Float64(g * g)) / 0.5)));
	else
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g / 2.0))));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-318], N[Power[t$95$0, 1/3], $MachinePrecision], If[LessEqual[t$95$0, 2e-319], N[(g / N[Power[N[(N[(a * N[(g * g), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(a / N[(g / 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.9999875e-319

   1. Initial program 88.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing

   if -9.9999875e-319 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.99998e-319

   1. Initial program 7.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \sqrt[3]{g \cdot \frac{1}{2 \cdot a}} \]

      2. cbrt-prod N/A

         \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \]

      3. pow1/3 N/A

         \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{1}{2 \cdot a}\right)}^{\color{blue}{\frac{1}{3}}} \]

      4. *-commutative N/A

         \[\leadsto {\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{g}} \]

      5. inv-pow N/A

         \[\leadsto {\left({\left(2 \cdot a\right)}^{-1}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{g} \]

      6. pow-pow N/A

         \[\leadsto {\left(2 \cdot a\right)}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{\color{blue}{g}} \]

      7. unpow-prod-down N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(-1 \cdot \frac{1}{3}\right)}\right) \cdot \sqrt[3]{\color{blue}{g}} \]

      8. metadata-eval N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\frac{-1}{3}}\right) \cdot \sqrt[3]{g} \]

      9. metadata-eval N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right) \cdot \sqrt[3]{g} \]

      10. pow-flip N/A

          \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{{a}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{g} \]

      11. pow1/3 N/A

          \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{\sqrt[3]{a}}\right) \cdot \sqrt[3]{g} \]

      12. div-inv N/A

          \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)}}{\sqrt[3]{a}} \cdot \sqrt[3]{\color{blue}{g}} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{3}\right)\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      18. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      19. cbrt-lowering-cbrt.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
   5. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)} \cdot {2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right)}^{2}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({2}^{\left(\frac{\frac{-1}{3}}{2}\right)}\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{-1}{3}}{2}\right)\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

      5. metadata-eval 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{6}\right), 2\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   6. Applied egg-rr 98.1%

      \[\leadsto \frac{\color{blue}{{\left({2}^{-0.16666666666666666}\right)}^{2}} \cdot \sqrt[3]{g}}{\sqrt[3]{a}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {\left({2}^{\frac{-1}{6}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \color{blue}{{\left({2}^{\frac{-1}{6}}\right)}^{2}} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot {\color{blue}{\left({2}^{\frac{-1}{6}}\right)}}^{2} \]

      4. cbrt-div N/A

         \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{g}}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

      5. pow1/3 N/A

         \[\leadsto \frac{1}{{\left(\frac{a}{g}\right)}^{\frac{1}{3}}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

      6. pow-flip N/A

         \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot {\color{blue}{\left({2}^{\frac{-1}{6}}\right)}}^{2} \]

      7. metadata-eval N/A

         \[\leadsto {\left(\frac{a}{g}\right)}^{\frac{-1}{3}} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

      8. metadata-eval N/A

         \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \cdot {\left({2}^{\color{blue}{\frac{-1}{6}}}\right)}^{2} \]

      9. pow-pow N/A

         \[\leadsto {\left(\frac{a}{g}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \cdot {2}^{\color{blue}{\left(\frac{-1}{6} \cdot 2\right)}} \]

      10. pow-prod-down N/A

          \[\leadsto {\left(\frac{a}{g} \cdot 2\right)}^{\color{blue}{\left(\frac{-1}{6} \cdot 2\right)}} \]

      11. metadata-eval N/A

          \[\leadsto {\left(\frac{a}{g} \cdot \frac{1}{\frac{1}{2}}\right)}^{\left(\frac{-1}{6} \cdot 2\right)} \]

      12. div-inv N/A

          \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\left(\color{blue}{\frac{-1}{6}} \cdot 2\right)} \]

      13. metadata-eval N/A

          \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\frac{-1}{3}} \]

      14. metadata-eval N/A

          \[\leadsto {\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{3}}\right)} \]

      15. pow-pow N/A

          \[\leadsto {\left({\left(\frac{\frac{a}{g}}{\frac{1}{2}}\right)}^{-1}\right)}^{\color{blue}{\frac{1}{3}}} \]

      16. inv-pow N/A

          \[\leadsto {\left(\frac{1}{\frac{\frac{a}{g}}{\frac{1}{2}}}\right)}^{\frac{1}{3}} \]

      17. clear-num N/A

          \[\leadsto {\left(\frac{\frac{1}{2}}{\frac{a}{g}}\right)}^{\frac{1}{3}} \]

      18. pow1/3 N/A

          \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{\frac{a}{g}}} \]

      19. associate-/r/ N/A

          \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a} \cdot g} \]

      20. associate-*l/ N/A

          \[\leadsto \sqrt[3]{\frac{\frac{1}{2} \cdot g}{a}} \]

      21. cbrt-div N/A

          \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot g}}{\color{blue}{\sqrt[3]{a}}} \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{2} \cdot g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

      23. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{2} \cdot g\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      25. cbrt-lowering-cbrt.f64 98.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   8. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}}} \]

   10. Applied egg-rr 34.2%

       \[\leadsto \color{blue}{\frac{g}{\sqrt[3]{\frac{\left(g \cdot g\right) \cdot a}{0.5}}}} \]

   if 1.99998e-319 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 87.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \sqrt[3]{g \cdot \frac{1}{2 \cdot a}} \]

      2. cbrt-prod N/A

         \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \]

      3. pow1/3 N/A

         \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{1}{2 \cdot a}\right)}^{\color{blue}{\frac{1}{3}}} \]

      4. *-commutative N/A

         \[\leadsto {\left(\frac{1}{2 \cdot a}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{g}} \]

      5. inv-pow N/A

         \[\leadsto {\left({\left(2 \cdot a\right)}^{-1}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{g} \]

      6. pow-pow N/A

         \[\leadsto {\left(2 \cdot a\right)}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{\color{blue}{g}} \]

      7. unpow-prod-down N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(-1 \cdot \frac{1}{3}\right)}\right) \cdot \sqrt[3]{\color{blue}{g}} \]

      8. metadata-eval N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\frac{-1}{3}}\right) \cdot \sqrt[3]{g} \]

      9. metadata-eval N/A

         \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot {a}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right) \cdot \sqrt[3]{g} \]

      10. pow-flip N/A

          \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{{a}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{g} \]

      11. pow1/3 N/A

          \[\leadsto \left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{\sqrt[3]{a}}\right) \cdot \sqrt[3]{g} \]

      12. div-inv N/A

          \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)}}{\sqrt[3]{a}} \cdot \sqrt[3]{\color{blue}{g}} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{{2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{3}\right)\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      18. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

      19. cbrt-lowering-cbrt.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt[3]{g} \cdot {2}^{\frac{-1}{3}}}{\sqrt[3]{\color{blue}{a}}} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\frac{{2}^{\frac{-1}{3}}}{\sqrt[3]{a}}} \]

      3. div-inv N/A

         \[\leadsto \sqrt[3]{g} \cdot \left({2}^{\frac{-1}{3}} \cdot \color{blue}{\frac{1}{\sqrt[3]{a}}}\right) \]

      4. pow1/3 N/A

         \[\leadsto \sqrt[3]{g} \cdot \left({2}^{\frac{-1}{3}} \cdot \frac{1}{{a}^{\color{blue}{\frac{1}{3}}}}\right) \]

      5. pow-flip N/A

         \[\leadsto \sqrt[3]{g} \cdot \left({2}^{\frac{-1}{3}} \cdot {a}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt[3]{g} \cdot \left({2}^{\frac{-1}{3}} \cdot {a}^{\frac{-1}{3}}\right) \]

      7. pow-prod-down N/A

         \[\leadsto \sqrt[3]{g} \cdot {\left(2 \cdot a\right)}^{\color{blue}{\frac{-1}{3}}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{\frac{-1}{3}} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt[3]{g} \cdot {\left(a \cdot \frac{1}{\frac{1}{2}}\right)}^{\frac{-1}{3}} \]

      10. div-inv N/A

          \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{a}{\frac{1}{2}}\right)}^{\frac{-1}{3}} \]

      11. metadata-eval N/A

          \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{a}{\frac{1}{2}}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{3}}\right)} \]

      12. pow-pow N/A

          \[\leadsto \sqrt[3]{g} \cdot {\left({\left(\frac{a}{\frac{1}{2}}\right)}^{-1}\right)}^{\color{blue}{\frac{1}{3}}} \]

      13. inv-pow N/A

          \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{1}{\frac{a}{\frac{1}{2}}}\right)}^{\frac{1}{3}} \]

      14. clear-num N/A

          \[\leadsto \sqrt[3]{g} \cdot {\left(\frac{\frac{1}{2}}{a}\right)}^{\frac{1}{3}} \]

      15. pow1/3 N/A

          \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}} \]

      16. cbrt-prod N/A

          \[\leadsto \sqrt[3]{g \cdot \frac{\frac{1}{2}}{a}} \]

      17. *-commutative N/A

          \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a} \cdot g} \]

      18. associate-*l/ N/A

          \[\leadsto \sqrt[3]{\frac{\frac{1}{2} \cdot g}{a}} \]

      19. clear-num N/A

          \[\leadsto \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2} \cdot g}}} \]

      20. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{\mathsf{neg}\left(-1\right)}{\frac{a}{\frac{1}{2} \cdot g}}} \]

      21. cbrt-div N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(-1\right)}}{\color{blue}{\sqrt[3]{\frac{a}{\frac{1}{2} \cdot g}}}} \]

      22. metadata-eval N/A

          \[\leadsto \frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\color{blue}{a}}{\frac{1}{2} \cdot g}}} \]

      23. metadata-eval N/A

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{a}{\frac{1}{2} \cdot g}}}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(-1\right)}{\sqrt[3]{\color{blue}{\frac{a}{\frac{1}{2} \cdot g}}}} \]

      25. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \color{blue}{\left(\sqrt[3]{\frac{a}{\frac{1}{2} \cdot g}}\right)}\right) \]

   6. Applied egg-rr 88.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{\frac{g}{2}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-319}:\\ \;\;\;\;\frac{g}{\sqrt[3]{\frac{a \cdot \left(g \cdot g\right)}{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\frac{g}{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a \cdot 2}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* a 2.0))))

C

double code(double g, double a) {
	return cbrt((g / (a * 2.0)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (a * 2.0)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(a * 2.0)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing

3. Final simplification 78.7%

   \[\leadsto \sqrt[3]{\frac{g}{a \cdot 2}} \]
4. Add Preprocessing

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g \cdot \frac{0.5}{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (* g (/ 0.5 a))))

C

double code(double g, double a) {
	return cbrt((g * (0.5 / a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g * (0.5 / a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g * Float64(0.5 / a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{2 \cdot a}{g}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{2 \cdot a} \cdot g\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

   6. metadata-eval 78.7%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

4. Applied egg-rr 78.7%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a} \cdot g}} \]

5. Final simplification 78.7%

   \[\leadsto \sqrt[3]{g \cdot \frac{0.5}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))