2-ancestry mixing, zero discriminant

Percentage Accurate: 75.9% → 98.6%

Time: 1.7s

Alternatives: 2

Speedup: N/A×

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: 75.9% 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: 98.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-0.5}\right) \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 26.5

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

4. Applied rewrites 26.5%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. flip3-+ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. unpow3 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

6. Applied rewrites 10.6%

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

7. Taylor expanded in a around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 76.1

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

9. Applied rewrites 76.1%

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

    1. lift-/.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. cbrt-div N/A

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

    4. lower-/.f64 N/A

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

    5. lower-cbrt.f64 N/A

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

    6. lower-cbrt.f64 98.6

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

11. Applied rewrites 98.6%

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

Alternative 2: 75.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-0.5}\right) \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 26.5

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

4. Applied rewrites 26.5%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. flip3-+ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. unpow3 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

6. Applied rewrites 10.6%

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

7. Taylor expanded in a around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 76.1

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

9. Applied rewrites 76.1%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))