2-ancestry mixing, zero discriminant

Percentage Accurate: 75.2% → 98.7%

Time: 9.1s

Alternatives: 12

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: 75.2% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. clear-num 73.2%

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

   2. cbrt-div 73.8%

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

   3. metadata-eval 73.8%

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

   4. associate-/l* 73.7%

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

4. Applied egg-rr 73.7%

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

   1. associate-*r/ 73.8%

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

   2. *-commutative 73.8%

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

   3. associate-/l* 73.8%

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

6. Simplified 73.8%

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

   1. cbrt-prod 98.6%

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

   2. clear-num 98.6%

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

   3. cbrt-div 98.6%

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

   4. metadata-eval 98.6%

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

   5. div-inv 98.6%

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

   6. metadata-eval 98.6%

      \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \frac{1}{\sqrt[3]{g \cdot \color{blue}{0.5}}}} \]

   7. div-inv 98.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot 0.5}}}} \]

   8. add-cube-cbrt 97.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right) \cdot \sqrt[3]{\sqrt[3]{a}}}}{\sqrt[3]{g \cdot 0.5}}} \]

   9. associate-/l* 97.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}}} \]

   10. pow2 97.7%

       \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{2}} \cdot \frac{\sqrt[3]{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}} \]

8. Applied egg-rr 97.7%

   \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{2} \cdot \frac{\sqrt[3]{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}}} \]
9. Step-by-step derivation

   1. associate-*r/ 97.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}}} \]

   2. unpow2 97.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right)} \cdot \sqrt[3]{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}} \]

   3. rem-3cbrt-lft 98.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{a}}}{\sqrt[3]{g \cdot 0.5}}} \]

10. Simplified 98.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot 0.5}}}} \]
11. Step-by-step derivation

    1. clear-num 98.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}}}} \]

    2. associate-/r/ 98.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sqrt[3]{g \cdot 0.5}} \cdot \sqrt[3]{a}}} \]

    3. metadata-eval 98.6%

       \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{g \cdot 0.5}} \cdot \sqrt[3]{a}} \]

    4. metadata-eval 98.6%

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

    5. div-inv 98.6%

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

    6. cbrt-div 98.6%

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

    7. clear-num 98.6%

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

    8. /-rgt-identity 98.6%

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

    9. clear-num 98.6%

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

    10. div-inv 98.7%

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

    11. add-sqr-sqrt 48.1%

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

    12. associate-/l* 48.2%

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

    13. pow1/3 45.7%

        \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{2}{g}\right)}^{0.3333333333333333}}} \cdot \frac{\sqrt{\sqrt[3]{\frac{2}{g}}}}{\frac{1}{\sqrt[3]{a}}}} \]

    14. sqrt-pow1 45.7%

        \[\leadsto \frac{1}{\color{blue}{{\left(\frac{2}{g}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \frac{\sqrt{\sqrt[3]{\frac{2}{g}}}}{\frac{1}{\sqrt[3]{a}}}} \]

    15. metadata-eval 45.7%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{\color{blue}{0.16666666666666666}} \cdot \frac{\sqrt{\sqrt[3]{\frac{2}{g}}}}{\frac{1}{\sqrt[3]{a}}}} \]

    16. pow1/3 44.9%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{\sqrt{\color{blue}{{\left(\frac{2}{g}\right)}^{0.3333333333333333}}}}{\frac{1}{\sqrt[3]{a}}}} \]

    17. sqrt-pow1 44.9%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{\color{blue}{{\left(\frac{2}{g}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{\frac{1}{\sqrt[3]{a}}}} \]

    18. metadata-eval 44.9%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{{\left(\frac{2}{g}\right)}^{\color{blue}{0.16666666666666666}}}{\frac{1}{\sqrt[3]{a}}}} \]

    19. add-cbrt-cube 44.9%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{{\left(\frac{2}{g}\right)}^{0.16666666666666666}}{\color{blue}{\sqrt[3]{\left(\frac{1}{\sqrt[3]{a}} \cdot \frac{1}{\sqrt[3]{a}}\right) \cdot \frac{1}{\sqrt[3]{a}}}}}} \]

    20. un-div-inv 44.9%

        \[\leadsto \frac{1}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{{\left(\frac{2}{g}\right)}^{0.16666666666666666}}{\sqrt[3]{\color{blue}{\frac{\frac{1}{\sqrt[3]{a}} \cdot \frac{1}{\sqrt[3]{a}}}{\sqrt[3]{a}}}}}} \]

12. Applied egg-rr 44.9%

    \[\leadsto \frac{1}{\color{blue}{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot \frac{{\left(\frac{2}{g}\right)}^{0.16666666666666666}}{\sqrt[3]{\frac{1}{a}}}}} \]
13. Step-by-step derivation

    1. associate-*r/ 44.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\frac{2}{g}\right)}^{0.16666666666666666} \cdot {\left(\frac{2}{g}\right)}^{0.16666666666666666}}{\sqrt[3]{\frac{1}{a}}}}} \]

    2. pow-sqr 44.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{{\left(\frac{2}{g}\right)}^{\left(2 \cdot 0.16666666666666666\right)}}}{\sqrt[3]{\frac{1}{a}}}} \]

    3. metadata-eval 44.9%

       \[\leadsto \frac{1}{\frac{{\left(\frac{2}{g}\right)}^{\color{blue}{0.3333333333333333}}}{\sqrt[3]{\frac{1}{a}}}} \]

    4. unpow1/3 98.8%

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

14. Simplified 98.8%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. pow1/3 37.9%

      \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{0.3333333333333333}} \]

   2. associate-/r* 37.9%

      \[\leadsto {\color{blue}{\left(\frac{\frac{g}{2}}{a}\right)}}^{0.3333333333333333} \]

   3. div-inv 37.9%

      \[\leadsto {\color{blue}{\left(\frac{g}{2} \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 26.1%

      \[\leadsto \color{blue}{{\left(\frac{g}{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]

   5. pow1/3 49.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   6. div-inv 49.7%

      \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   7. metadata-eval 49.7%

      \[\leadsto \sqrt[3]{g \cdot \color{blue}{0.5}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.7%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\sqrt[3]{\frac{1}{a}}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot \sqrt[3]{\frac{1}{a}}} \]
7. Step-by-step derivation

   1. cbrt-div 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{a}}} \]

   2. metadata-eval 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \frac{\color{blue}{1}}{\sqrt[3]{a}} \]

   3. div-inv 98.6%

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

   4. clear-num 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot 0.5}}}} \]

   5. div-inv 98.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{a} \cdot \frac{1}{\sqrt[3]{g \cdot 0.5}}}} \]

   6. metadata-eval 98.6%

      \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{g \cdot 0.5}}} \]

   7. metadata-eval 98.6%

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

   8. div-inv 98.6%

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

   9. cbrt-div 98.6%

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

   10. clear-num 98.6%

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

   11. associate-/r* 98.7%

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

8. Applied egg-rr 98.7%

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

   1. clear-num 98.7%

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

   2. associate-/r/ 98.7%

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

   3. add-cbrt-cube 98.5%

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

   4. un-div-inv 98.5%

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

   5. frac-times 98.4%

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

   6. metadata-eval 98.4%

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

   7. associate-/r* 98.4%

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

   8. add-cube-cbrt 98.8%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

    \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{2}{g}}} \]
12. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. pow1/3 37.9%

      \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{0.3333333333333333}} \]

   2. associate-/r* 37.9%

      \[\leadsto {\color{blue}{\left(\frac{\frac{g}{2}}{a}\right)}}^{0.3333333333333333} \]

   3. div-inv 37.9%

      \[\leadsto {\color{blue}{\left(\frac{g}{2} \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 26.1%

      \[\leadsto \color{blue}{{\left(\frac{g}{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]

   5. pow1/3 49.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   6. div-inv 49.7%

      \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   7. metadata-eval 49.7%

      \[\leadsto \sqrt[3]{g \cdot \color{blue}{0.5}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.7%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\sqrt[3]{\frac{1}{a}}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot \sqrt[3]{\frac{1}{a}}} \]
7. Step-by-step derivation

   1. cbrt-div 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{a}}} \]

   2. metadata-eval 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \frac{\color{blue}{1}}{\sqrt[3]{a}} \]

   3. div-inv 98.6%

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

   4. clear-num 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot 0.5}}}} \]

   5. div-inv 98.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{a} \cdot \frac{1}{\sqrt[3]{g \cdot 0.5}}}} \]

   6. metadata-eval 98.6%

      \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{g \cdot 0.5}}} \]

   7. metadata-eval 98.6%

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

   8. div-inv 98.6%

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

   9. cbrt-div 98.6%

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

   10. clear-num 98.6%

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

   11. associate-/r* 98.7%

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in a around 0 98.7%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. pow1/3 37.9%

      \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{0.3333333333333333}} \]

   2. associate-/r* 37.9%

      \[\leadsto {\color{blue}{\left(\frac{\frac{g}{2}}{a}\right)}}^{0.3333333333333333} \]

   3. div-inv 37.9%

      \[\leadsto {\color{blue}{\left(\frac{g}{2} \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 26.1%

      \[\leadsto \color{blue}{{\left(\frac{g}{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]

   5. pow1/3 49.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   6. div-inv 49.7%

      \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   7. metadata-eval 49.7%

      \[\leadsto \sqrt[3]{g \cdot \color{blue}{0.5}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.7%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\sqrt[3]{\frac{1}{a}}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot \sqrt[3]{\frac{1}{a}}} \]

7. Final simplification 98.7%

   \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot 0.5} \]
8. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. associate-/r* 73.4%

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

   2. cbrt-div 98.6%

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

   3. div-inv 98.6%

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

   4. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. cbrt-div 98.6%

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

   2. clear-num 98.6%

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

4. Applied egg-rr 98.6%

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

   1. associate-/r/ 98.6%

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

   2. associate-*l/ 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 7: 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 73.4%

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

   1. pow1/3 37.9%

      \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{0.3333333333333333}} \]

   2. clear-num 37.8%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{2 \cdot a}{g}}\right)}}^{0.3333333333333333} \]

   3. associate-/r/ 37.9%

      \[\leadsto {\color{blue}{\left(\frac{1}{2 \cdot a} \cdot g\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 26.1%

      \[\leadsto \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{0.3333333333333333} \cdot {g}^{0.3333333333333333}} \]

   5. pow1/3 44.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}}} \cdot {g}^{0.3333333333333333} \]

   6. associate-/r* 44.9%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot {g}^{0.3333333333333333} \]

   7. metadata-eval 44.9%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a}} \cdot {g}^{0.3333333333333333} \]

   8. pow1/3 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 8: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. clear-num 73.2%

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

   2. cbrt-div 73.8%

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

   3. metadata-eval 73.8%

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

   4. associate-/l* 73.7%

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

4. Applied egg-rr 73.7%

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

   1. associate-*r/ 73.8%

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

   2. *-commutative 73.8%

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

   3. associate-/l* 73.8%

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

6. Simplified 73.8%

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

   1. associate-*r/ 73.8%

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

8. Applied egg-rr 73.8%

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

9. Final simplification 73.8%

   \[\leadsto \frac{1}{\sqrt[3]{\frac{2 \cdot a}{g}}} \]
10. Add Preprocessing

Alternative 9: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. clear-num 73.2%

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

   2. cbrt-div 73.8%

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

   3. metadata-eval 73.8%

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

   4. associate-/l* 73.7%

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

4. Applied egg-rr 73.7%

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

   1. associate-*r/ 73.8%

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

   2. *-commutative 73.8%

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

   3. associate-/l* 73.8%

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

6. Simplified 73.8%

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

7. Final simplification 73.8%

   \[\leadsto \frac{1}{\sqrt[3]{\frac{2}{g} \cdot a}} \]
8. Add Preprocessing

Alternative 10: 75.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. clear-num 73.2%

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

   2. cbrt-div 73.8%

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

   3. metadata-eval 73.8%

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

   4. associate-/l* 73.7%

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

4. Applied egg-rr 73.7%

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

Alternative 11: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. pow1/3 37.9%

      \[\leadsto \color{blue}{{\left(\frac{g}{2 \cdot a}\right)}^{0.3333333333333333}} \]

   2. associate-/r* 37.9%

      \[\leadsto {\color{blue}{\left(\frac{\frac{g}{2}}{a}\right)}}^{0.3333333333333333} \]

   3. div-inv 37.9%

      \[\leadsto {\color{blue}{\left(\frac{g}{2} \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \]

   4. unpow-prod-down 26.1%

      \[\leadsto \color{blue}{{\left(\frac{g}{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]

   5. pow1/3 49.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   6. div-inv 49.7%

      \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

   7. metadata-eval 49.7%

      \[\leadsto \sqrt[3]{g \cdot \color{blue}{0.5}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.7%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\sqrt[3]{\frac{1}{a}}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot \sqrt[3]{\frac{1}{a}}} \]
7. Step-by-step derivation

   1. cbrt-div 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{a}}} \]

   2. metadata-eval 98.6%

      \[\leadsto \sqrt[3]{g \cdot 0.5} \cdot \frac{\color{blue}{1}}{\sqrt[3]{a}} \]

   3. div-inv 98.6%

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

   4. clear-num 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot 0.5}}}} \]

   5. div-inv 98.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{a} \cdot \frac{1}{\sqrt[3]{g \cdot 0.5}}}} \]

   6. metadata-eval 98.6%

      \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{g \cdot 0.5}}} \]

   7. metadata-eval 98.6%

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

   8. div-inv 98.6%

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

   9. cbrt-div 98.6%

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

   10. clear-num 98.6%

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

   11. associate-/r* 98.7%

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

8. Applied egg-rr 98.7%

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

   1. add-cbrt-cube 98.5%

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

   2. cbrt-undiv 73.2%

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

   3. un-div-inv 73.2%

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

   4. frac-times 73.2%

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

   5. metadata-eval 73.2%

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

   6. associate-/r* 73.3%

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

   7. add-cube-cbrt 73.4%

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

10. Applied egg-rr 73.4%

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

Alternative 12: 75.2% 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 73.4%

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

   1. add-log-exp 9.8%

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

   2. *-un-lft-identity 9.8%

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

   3. log-prod 9.8%

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

   4. metadata-eval 9.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt[3]{\frac{g}{2 \cdot a}}}\right) \]

   5. add-log-exp 73.4%

      \[\leadsto 0 + \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

   6. div-inv 73.4%

      \[\leadsto 0 + \sqrt[3]{\color{blue}{g \cdot \frac{1}{2 \cdot a}}} \]

   7. associate-/r* 73.4%

      \[\leadsto 0 + \sqrt[3]{g \cdot \color{blue}{\frac{\frac{1}{2}}{a}}} \]

   8. metadata-eval 73.4%

      \[\leadsto 0 + \sqrt[3]{g \cdot \frac{\color{blue}{0.5}}{a}} \]

4. Applied egg-rr 73.4%

   \[\leadsto \color{blue}{0 + \sqrt[3]{g \cdot \frac{0.5}{a}}} \]
5. Step-by-step derivation

   1. +-lft-identity 73.4%

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

6. Simplified 73.4%

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

Reproduce

herbie shell --seed 2024093 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))