2-ancestry mixing, zero discriminant

Percentage Accurate: 76.5% → 98.7%

Time: 8.4s

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

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

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 74.3%

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

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

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

   2. associate-/l/ N/A

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

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

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

   4. /-lowering-/.f64 74.3%

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

3. Simplified 74.3%

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

   1. associate-/l/ N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

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

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

   5. pow1/3 N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

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

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

Alternative 2: 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. Step-by-step derivation

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

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

   2. associate-/l/ N/A

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

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

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

   4. /-lowering-/.f64 74.3%

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

3. Simplified 74.3%

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

   1. associate-/l/ N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

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

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

   5. pow1/3 N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

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

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

   1. cbrt-undiv N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. cbrt-div N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

   10. cbrt-lowering-cbrt.f64 98.8%

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

8. Applied egg-rr 98.8%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

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

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. un-div-inv N/A

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

   13. /-lowering-/.f64 98.8%

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

10. Applied egg-rr 98.8%

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

    1. cbrt-div N/A

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

    2. metadata-eval N/A

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

    3. associate-/r* N/A

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

    4. associate-/l/ N/A

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

    5. cbrt-div N/A

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

    6. associate-/r/ N/A

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

    7. associate-/r* N/A

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

    8. cbrt-prod N/A

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

    9. metadata-eval N/A

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

    10. div-inv N/A

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

    11. pow1/3 N/A

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

    12. pow-flip N/A

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

    13. metadata-eval N/A

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

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

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

    15. clear-num N/A

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

    16. inv-pow N/A

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

    17. pow-pow N/A

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

    18. metadata-eval N/A

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

    19. pow1/3 N/A

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

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

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

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

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

    22. cbrt-lowering-cbrt.f64 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 3: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{\frac{a}{0.5}}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (cbrt (/ (/ a 0.5) g))))

C

double code(double g, double a) {
	return 1.0 / cbrt(((a / 0.5) / g));
}

Java

public static double code(double g, double a) {
	return 1.0 / Math.cbrt(((a / 0.5) / g));
}

Julia

function code(g, a)
	return Float64(1.0 / cbrt(Float64(Float64(a / 0.5) / g)))
end

Wolfram

code[g_, a_] := N[(1.0 / N[Power[N[(N[(a / 0.5), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

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

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

   2. associate-/l/ N/A

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

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

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

   4. /-lowering-/.f64 74.3%

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

3. Simplified 74.3%

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

   1. associate-/l/ N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

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

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

   5. pow1/3 N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

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

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

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

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

   17. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

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

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

   3. cbrt-undiv N/A

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

   4. *-lft-identity N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. metadata-eval N/A

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

   8. *-inverses N/A

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

   9. div-inv N/A

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

   10. associate-*r* N/A

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

   11. associate-/r/ N/A

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

   12. clear-num N/A

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

   13. frac-times N/A

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

   14. frac-2neg N/A

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

   15. distribute-neg-frac2 N/A

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

   16. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{2}{a} \cdot \left(\mathsf{neg}\left(\frac{0 - a}{\frac{g}{a}}\right)\right)}\right)\right) \]

   17. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{2}{\frac{a}{\mathsf{neg}\left(\frac{0 - a}{\frac{g}{a}}\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{2}{\frac{a}{\mathsf{neg}\left(\frac{0 - a}{\frac{g}{a}}\right)}}\right)\right)\right) \]

   19. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{2}{a} \cdot \left(\mathsf{neg}\left(\frac{0 - a}{\frac{g}{a}}\right)\right)\right)\right)\right) \]

   20. sub0-neg N/A

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

8. Applied egg-rr 74.9%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{\frac{a}{0.5}}{g}}}} \]
9. 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 74.3%

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

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

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

   2. associate-/l/ N/A

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

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

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

   4. /-lowering-/.f64 74.3%

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

3. Simplified 74.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-/l/ N/A

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

   6. associate-/r* N/A

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

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

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

   8. metadata-eval 74.3%

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

6. Applied egg-rr 74.3%

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

7. Final simplification 74.3%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))