2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 5.8s

Alternatives: 7

Speedup: 0.5×

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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. cbrt-div N/A

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

   5. neg-mul-1 N/A

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

   6. cbrt-prod N/A

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

   7. pow1/3 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/3 N/A

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

   11. sqr-pow N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow-prod-down N/A

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

   17. sqr-pow N/A

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

   18. pow1/3 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. pow1/3 N/A

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

   22. lower-cbrt.f64 N/A

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

   23. lower-cbrt.f64 1.7

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

   24. remove-double-div N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-prod N/A

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

   9. neg-mul-1 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval 98.8

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

8. Applied rewrites 98.8%

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

Alternative 2: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt[3]{-0.5 \cdot g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-307)
   (* (cbrt (* -0.5 g)) (pow (- a) -0.3333333333333333))
   (* (cbrt (* 0.5 g)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-307) {
		tmp = cbrt((-0.5 * g)) * pow(-a, -0.3333333333333333);
	} else {
		tmp = cbrt((0.5 * g)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-307) {
		tmp = Math.cbrt((-0.5 * g)) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.cbrt((0.5 * g)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-307)
		tmp = Float64(cbrt(Float64(-0.5 * g)) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = Float64(cbrt(Float64(0.5 * g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-307], N[(N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -3.99999999999999964e-307

   1. Initial program 76.3%

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

   4. Applied rewrites 92.1%

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

   if -3.99999999999999964e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. cbrt-div N/A

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

      5. neg-mul-1 N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/3 N/A

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

      11. sqr-pow N/A

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

      12. pow-prod-down N/A

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

      13. sqr-neg N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. pow-prod-down N/A

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

      17. sqr-pow N/A

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

      18. pow1/3 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-cbrt.f64 N/A

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

      21. pow1/3 N/A

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

      22. lower-cbrt.f64 N/A

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

      23. lower-cbrt.f64 1.7

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

      24. remove-double-div N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. cbrt-undiv N/A

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

      9. cbrt-undiv N/A

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

      10. lift-*.f64 N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-/l* N/A

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

      17. associate-*l/ N/A

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

      18. lift-/.f64 N/A

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

      19. cbrt-div N/A

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

      20. associate-/l/ N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 92.0%

      \[\leadsto \color{blue}{{a}^{-0.3333333333333333} \cdot \sqrt[3]{0.5 \cdot g}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt[3]{-0.5 \cdot g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))