2-ancestry mixing, zero discriminant

Percentage Accurate: 75.0% → 98.7%

Time: 4.9s

Alternatives: 6

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: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

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 77.6%

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

   1. expm1-log1p-u 58.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 26.6%

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

   4. add-exp-log 45.5%

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

   5. *-un-lft-identity 45.5%

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

   6. times-frac 45.5%

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

   7. metadata-eval 45.5%

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

3. Applied egg-rr 45.5%

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

   1. +-commutative 45.5%

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

   2. associate--l+ 77.6%

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

   3. metadata-eval 77.6%

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

   4. +-rgt-identity 77.6%

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

   5. associate-*r/ 77.6%

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

   6. associate-*l/ 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in a around 0 77.6%

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

   1. associate-*r/ 77.6%

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

   2. associate-/l* 77.0%

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

8. Simplified 77.0%

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

   1. cbrt-div 77.2%

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

   2. clear-num 77.1%

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

   3. cbrt-div 98.6%

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

   4. associate-/r* 98.6%

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

   5. cbrt-prod 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

   8. metadata-eval 98.6%

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

   9. div-inv 98.6%

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

   10. cbrt-div 98.7%

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

   11. clear-num 98.7%

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

   12. /-rgt-identity 98.7%

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

   13. clear-num 98.6%

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

   14. div-inv 98.6%

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

   15. clear-num 98.7%

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

   16. associate-/l/ 98.7%

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

   17. associate-/r* 98.8%

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

10. Applied egg-rr 98.8%

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

    1. add-cbrt-cube 98.6%

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

    2. un-div-inv 98.5%

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

    3. frac-times 98.5%

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

    4. metadata-eval 98.5%

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

    5. associate-/r* 98.6%

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

    6. add-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Final simplification 98.8%

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

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 77.6%

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

   1. div-inv 77.6%

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

   2. cbrt-prod 98.7%

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

   3. associate-/r* 98.7%

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

   4. metadata-eval 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. cbrt-div 98.7%

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

   2. div-inv 98.6%

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

3. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

   3. *-commutative 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. expm1-log1p-u 58.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 26.6%

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

   4. add-exp-log 45.5%

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

   5. *-un-lft-identity 45.5%

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

   6. times-frac 45.5%

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

   7. metadata-eval 45.5%

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

3. Applied egg-rr 45.5%

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

   1. +-commutative 45.5%

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

   2. associate--l+ 77.6%

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

   3. metadata-eval 77.6%

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

   4. +-rgt-identity 77.6%

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

   5. associate-*r/ 77.6%

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

   6. associate-*l/ 77.6%

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

5. Simplified 77.6%

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

6. Final simplification 77.6%

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

Alternative 5: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Final simplification 77.6%

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

Alternative 6: 3.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g} \cdot 0 \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt g) 0.0))

C

double code(double g, double a) {
	return cbrt(g) * 0.0;
}

Java

public static double code(double g, double a) {
	return Math.cbrt(g) * 0.0;
}

Julia

function code(g, a)
	return Float64(cbrt(g) * 0.0)
end

Wolfram

code[g_, a_] := N[(N[Power[g, 1/3], $MachinePrecision] * 0.0), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

   1. expm1-log1p-u 58.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 26.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 26.6%

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

   4. add-exp-log 45.5%

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

   5. *-un-lft-identity 45.5%

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

   6. times-frac 45.5%

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

   7. metadata-eval 45.5%

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

3. Applied egg-rr 45.5%

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

   1. +-commutative 45.5%

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

   2. associate--l+ 77.6%

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

   3. metadata-eval 77.6%

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

   4. +-rgt-identity 77.6%

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

   5. associate-*r/ 77.6%

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

   6. associate-*l/ 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in a around 0 77.6%

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

   1. associate-*r/ 77.6%

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

   2. associate-/l* 77.0%

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

8. Simplified 77.0%

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

   1. cbrt-div 77.2%

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

   2. clear-num 77.1%

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

   3. cbrt-div 98.6%

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

   4. associate-/r* 98.6%

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

   5. cbrt-prod 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

   8. metadata-eval 98.6%

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

   9. div-inv 98.6%

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

   10. cbrt-div 98.7%

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

   11. clear-num 98.7%

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

   12. /-rgt-identity 98.7%

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

   13. clear-num 98.6%

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

   14. div-inv 98.6%

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

   15. clear-num 98.7%

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

   16. associate-/l/ 98.7%

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

   17. associate-/r* 98.8%

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

10. Applied egg-rr 98.8%

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

    1. associate-/l/ 98.7%

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

    2. inv-pow 98.7%

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

12. Applied egg-rr 6.2%

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

14. Simplified 3.2%

    \[\leadsto \color{blue}{0 \cdot \sqrt[3]{g}} \]

15. Final simplification 3.2%

    \[\leadsto \sqrt[3]{g} \cdot 0 \]

Reproduce

herbie shell --seed 2023213 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))