2-ancestry mixing, zero discriminant

Percentage Accurate: 76.5% → 98.7%

Time: 8.2s

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: 76.5% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.7%

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

6. Applied egg-rr 98.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot {\left({2}^{\frac{-1}{6}}\right)}^{2}\right), \mathsf{cbrt.f64}\left(\color{blue}{a}\right)\right) \]

   2. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot {2}^{\left(\frac{-1}{6} \cdot 2\right)}\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. div-inv N/A

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

   8. cbrt-div N/A

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

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

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

   10. /-lowering-/.f64 98.8%

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

8. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]
9. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.7%

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

6. Applied egg-rr 98.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot {\left({2}^{\frac{-1}{6}}\right)}^{2}\right), \mathsf{cbrt.f64}\left(\color{blue}{a}\right)\right) \]

   2. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot {2}^{\left(\frac{-1}{6} \cdot 2\right)}\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. div-inv N/A

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

   8. cbrt-div N/A

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

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

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

   10. /-lowering-/.f64 98.8%

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

8. Applied egg-rr 98.8%

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

   1. cbrt-undiv N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-/l* N/A

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

   5. cbrt-unprod N/A

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

   6. cbrt-undiv N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. /-lowering-/.f64 N/A

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

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

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

   11. cbrt-undiv N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac2 N/A

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

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

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

   15. distribute-neg-frac2 N/A

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

   16. metadata-eval N/A

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

   17. /-lowering-/.f64 98.8%

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

10. Applied egg-rr 98.8%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. *-lowering-*.f64 N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

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

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

   9. remove-double-div N/A

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

   10. associate-/r* N/A

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

   11. clear-num N/A

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

   12. /-lowering-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. cbrt-lowering-cbrt.f64 47.4%

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

4. Applied egg-rr 47.4%

   \[\leadsto \color{blue}{{\left(\frac{a}{0.5}\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}} \]
5. Step-by-step derivation

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow1/3 N/A

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

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

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

   7. /-lowering-/.f64 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 76.5% 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 77.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 77.5%

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

4. Applied egg-rr 77.5%

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

5. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))