2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 8.7s

Alternatives: 5

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: 76.4% 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

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{\frac{2}{\frac{1}{a}}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt g) (cbrt (/ 2.0 (/ 1.0 a)))))

C

double code(double g, double a) {
	return cbrt(g) / cbrt((2.0 / (1.0 / a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) / Math.cbrt((2.0 / (1.0 / a)));
}

Julia

function code(g, a)
	return Float64(cbrt(g) / cbrt(Float64(2.0 / Float64(1.0 / a))))
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] / N[Power[N[(2.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

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

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

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

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

   8. distribute-neg-frac2 N/A

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

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

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. /-lowering-/.f64 46.4%

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

4. Applied egg-rr 46.4%

   \[\leadsto \color{blue}{{\left(\frac{g}{-2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\frac{-1}{a}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

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

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

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

   3. /-lowering-/.f64 98.8%

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

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{-2}}} \cdot \sqrt[3]{\frac{-1}{a}} \]
7. Step-by-step derivation

   1. cbrt-unprod N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. frac-times N/A

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

   5. cbrt-div N/A

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

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

   8. remove-double-neg N/A

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

   9. pow1/3 N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. frac-2neg N/A

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

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

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

   19. pow1/3 N/A

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

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

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

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

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

   22. /-lowering-/.f64 98.8%

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

8. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{\frac{a}{0.5}}}} \]
9. Step-by-step derivation

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

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

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

   6. /-lowering-/.f64 98.8%

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

10. Applied egg-rr 98.8%

    \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\frac{2}{\frac{1}{a}}}}} \]
11. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{-2}} \cdot \sqrt[3]{\frac{-1}{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt (/ g -2.0)) (cbrt (/ -1.0 a))))

C

double code(double g, double a) {
	return cbrt((g / -2.0)) * cbrt((-1.0 / a));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / -2.0)) * Math.cbrt((-1.0 / a));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g / -2.0)) * cbrt(Float64(-1.0 / a)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g / -2.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

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

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

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

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

   8. distribute-neg-frac2 N/A

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

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

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. /-lowering-/.f64 46.4%

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

4. Applied egg-rr 46.4%

   \[\leadsto \color{blue}{{\left(\frac{g}{-2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\frac{-1}{a}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

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

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

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

   3. /-lowering-/.f64 98.8%

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

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{-2}}} \cdot \sqrt[3]{\frac{-1}{a}} \]
7. 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 74.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 74.3%

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

4. Applied egg-rr 74.3%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a} \cdot g}} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. frac-2neg N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. distribute-frac-neg2 N/A

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

   8. clear-num N/A

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

   9. *-inverses N/A

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

   10. un-div-inv N/A

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

   11. clear-num N/A

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

   12. associate-/l/ N/A

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

   13. times-frac N/A

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

   14. *-commutative N/A

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

   15. distribute-frac-neg2 N/A

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

   16. div-inv N/A

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

   17. associate-*l* N/A

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

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

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

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

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

   20. metadata-eval N/A

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

   21. frac-2neg N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. remove-double-neg N/A

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

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

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

6. Applied egg-rr 54.8%

   \[\leadsto \sqrt[3]{\color{blue}{g \cdot \left(g \cdot \frac{-1}{a \cdot \left(g \cdot -2\right)}\right)}} \]
7. Step-by-step derivation

   1. pow1/3 N/A

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

   2. *-commutative N/A

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

   3. unpow-prod-down N/A

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

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

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

   5. pow1/3 N/A

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

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

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. un-div-inv N/A

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

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

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

   11. distribute-rgt-neg-in N/A

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

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

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

   13. distribute-rgt-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

   16. pow1/3 N/A

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

   17. cbrt-lowering-cbrt.f64 79.0%

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

8. Applied egg-rr 79.0%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. cbrt-div N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. un-div-inv N/A

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

   7. rgt-mult-inverse N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. div-inv N/A

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

   11. div-inv N/A

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

   12. clear-num N/A

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

   13. clear-num N/A

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

   14. cbrt-div N/A

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

   15. clear-num N/A

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

   16. clear-num N/A

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

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

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

   18. /-lowering-/.f64 98.8%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 4: 76.4% 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]

Derivation

1. Initial program 74.3%

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

Alternative 5: 76.4% 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 74.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 74.3%

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

4. Applied egg-rr 74.3%

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

5. Final simplification 74.3%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))