2-ancestry mixing, zero discriminant

Percentage Accurate: 75.6% → 98.7%

Time: 4.4s

Alternatives: 11

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: 75.6% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-cbrt.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. un-div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. lift-/.f64 N/A

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

   17. unpow-prod-down N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. pow1/3 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. pow1/3 N/A

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

   23. lower-cbrt.f64 98.8

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{\frac{-1}{\frac{-1}{a}} \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{0.5 \cdot g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-305)
   (cbrt (/ (- g) (* (/ -1.0 (/ -1.0 a)) -2.0)))
   (* (pow a -0.3333333333333333) (cbrt (* 0.5 g)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-305) {
		tmp = cbrt((-g / ((-1.0 / (-1.0 / a)) * -2.0)));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((0.5 * g));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-305) {
		tmp = Math.cbrt((-g / ((-1.0 / (-1.0 / a)) * -2.0)));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((0.5 * g));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-305)
		tmp = cbrt(Float64(Float64(-g) / Float64(Float64(-1.0 / Float64(-1.0 / a)) * -2.0)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(0.5 * g)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-305], N[Power[N[((-g) / N[(N[(-1.0 / N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.99999999999999985e-305

   1. Initial program 68.5%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. flip-+ N/A

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

      4. +-inverses N/A

         \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{0}}{a - a}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{g}{\frac{\color{blue}{\mathsf{neg}\left(0\right)}}{a - a}}} \]

      6. +-inverses N/A

         \[\leadsto \sqrt[3]{\frac{g}{\frac{\mathsf{neg}\left(0\right)}{\color{blue}{0}}}} \]

      7. distribute-neg-frac N/A

         \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\mathsf{neg}\left(\frac{0}{0}\right)}}} \]

      8. +-inverses N/A

         \[\leadsto \sqrt[3]{\frac{g}{\mathsf{neg}\left(\frac{\color{blue}{a \cdot a - a \cdot a}}{0}\right)}} \]

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. count-2-rev N/A

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

      12. lift-*.f64 N/A

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

      13. neg-sub0 N/A

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

      14. flip-- N/A

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. flip-+ N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \frac{\color{blue}{\mathsf{neg}\left(0\right)}}{a - a}}}} \]

      20. +-inverses N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \frac{\mathsf{neg}\left(0\right)}{\color{blue}{0}}}}} \]

      21. distribute-neg-frac N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \color{blue}{\left(\mathsf{neg}\left(\frac{0}{0}\right)\right)}}}} \]

      22. +-inverses N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \left(\mathsf{neg}\left(\frac{\color{blue}{a \cdot a - a \cdot a}}{0}\right)\right)}}} \]

      23. +-inverses N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \left(\mathsf{neg}\left(\frac{a \cdot a - a \cdot a}{\color{blue}{a - a}}\right)\right)}}} \]

      24. flip-+ N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \left(\mathsf{neg}\left(\color{blue}{\left(a + a\right)}\right)\right)}}} \]

      25. count-2-rev N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \left(\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)\right)}}} \]

      26. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{0 + \left(\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)\right)}}} \]

      27. sub-neg N/A

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

      28. neg-sub0 N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\color{blue}{\mathsf{neg}\left(2 \cdot a\right)}}}} \]

      29. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)}}} \]

      30. count-2-rev N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\mathsf{neg}\left(\color{blue}{\left(a + a\right)}\right)}}} \]

      31. flip-+ N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\mathsf{neg}\left(\color{blue}{\frac{a \cdot a - a \cdot a}{a - a}}\right)}}} \]

      32. +-inverses N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\mathsf{neg}\left(\frac{\color{blue}{0}}{a - a}\right)}}} \]

      33. +-inverses N/A

          \[\leadsto \sqrt[3]{\frac{g}{\frac{0 \cdot 0 - \left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)}}} \]

   4. Applied rewrites 40.8%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}{-2 \cdot a}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\frac{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}{-2 \cdot a}\right)}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{-g}}{\mathsf{neg}\left(\frac{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}{-2 \cdot a}\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\frac{-g}{\mathsf{neg}\left(\color{blue}{\frac{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}{-2 \cdot a}}\right)}} \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt[3]{\frac{-g}{\color{blue}{\frac{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}{\mathsf{neg}\left(-2 \cdot a\right)}}}} \]

      6. lift--.f64 N/A

         \[\leadsto \sqrt[3]{\frac{-g}{\frac{\color{blue}{0 - \left(-2 \cdot a\right) \cdot \left(-2 \cdot a\right)}}{\mathsf{neg}\left(-2 \cdot a\right)}}} \]

      7. sub0-neg N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{-g}{\frac{\color{blue}{4}}{-2} \cdot \frac{a \cdot a}{a}}} \]

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. mul-1-neg N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 68.5

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

   8. Applied rewrites 68.5%

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

   if -4.99999999999999985e-305 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 71.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. cbrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-cbrt.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 92.0

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

   6. Applied rewrites 92.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt[3]{\frac{-g}{\frac{-1}{\frac{-1}{a}} \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{0.5 \cdot g}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

Alternative 4: 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 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-cbrt.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. un-div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. lift-/.f64 N/A

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

   17. unpow-prod-down N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. pow1/3 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. pow1/3 N/A

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

   23. lower-cbrt.f64 98.8

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. unpow-prod-down N/A

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

   11. lift-/.f64 N/A

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

   12. times-frac N/A

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

   13. neg-mul-1 N/A

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

   14. lift-neg.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

8. Applied rewrites 98.6%

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

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-cbrt.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. un-div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. lift-/.f64 N/A

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

   17. unpow-prod-down N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. pow1/3 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. pow1/3 N/A

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

   23. lower-cbrt.f64 98.8

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 98.6%

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

Alternative 6: 75.5% accurate, 0.9× 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 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-cbrt.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. un-div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. lift-/.f64 N/A

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

   17. unpow-prod-down N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. pow1/3 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. pow1/3 N/A

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

   23. lower-cbrt.f64 98.8

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 70.3%

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

Alternative 7: 75.4% accurate, 0.9× 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 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lift-*.f64 N/A

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

   6. count-2-rev N/A

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

   7. flip-+ N/A

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

   8. +-inverses N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{\color{blue}{0}}{a - a}\right)}} \]

   9. +-inverses N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)}} \]

   10. distribute-neg-frac N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{0}}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\frac{\color{blue}{0}}{0}}} \]

   12. +-inverses N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\frac{\color{blue}{a \cdot a - a \cdot a}}{0}}} \]

   13. +-inverses N/A

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

   14. flip-+ N/A

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

   15. count-2-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-cbrt.f64 N/A

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

   19. lower-neg.f64 N/A

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

   20. lower-cbrt.f64 1.7

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

   21. lift-*.f64 N/A

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

   22. count-2-rev N/A

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

   23. flip-+ N/A

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

   24. +-inverses N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\frac{\color{blue}{0}}{a - a}}} \]

   25. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\frac{\color{blue}{\mathsf{neg}\left(0\right)}}{a - a}}} \]

   26. +-inverses N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\frac{\mathsf{neg}\left(0\right)}{\color{blue}{0}}}} \]

   27. distribute-neg-frac N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\color{blue}{\mathsf{neg}\left(\frac{0}{0}\right)}}} \]

   28. +-inverses N/A

       \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{\mathsf{neg}\left(\frac{\color{blue}{a \cdot a - a \cdot a}}{0}\right)}} \]

   29. +-inverses N/A

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

4. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. cbrt-undiv N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 70.3

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

6. Applied rewrites 70.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. cbrt-prod N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lift-neg.f64 N/A

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

   11. remove-double-neg N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f64 N/A

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

   14. cbrt-prod N/A

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

   15. div-inv N/A

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

   16. cbrt-undiv N/A

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

   17. lift-cbrt.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. div-inv N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

8. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-*.f64 70.3

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

10. Applied rewrites 70.3%

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

Alternative 8: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. times-frac N/A

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

   8. distribute-neg-frac2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. div-inv N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 70.0

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

4. Applied rewrites 70.0%

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

5. Final simplification 70.0%

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

Alternative 9: 75.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (+ a a))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. lower-+.f64 69.9

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

4. Applied rewrites 69.9%

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

Alternative 10: 5.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(2 \cdot a\right) \cdot g} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. pow1/3 N/A

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

   3. lower-pow.f64 33.2

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

4. Applied rewrites 1.2%

   \[\leadsto \color{blue}{{\left(\left(-2 \cdot a\right) \cdot g\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow1/3 N/A

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

   3. lower-cbrt.f64 1.7

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

   4. lift-*.f64 N/A

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

   5. /-rgt-identity N/A

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

   6. clear-num N/A

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

   7. unpow-1 N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. pow2 N/A

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

   11. sqr-neg-rev N/A

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

   12. pow-prod-down N/A

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

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

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

   15. inv-pow N/A

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

   16. clear-num-rev N/A

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

   17. metadata-eval N/A

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

   18. frac-2neg N/A

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

   19. associate-/l* N/A

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

   20. *-commutative N/A

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

   21. associate-/l* N/A

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

   22. metadata-eval N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 5.7

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

6. Applied rewrites 5.7%

   \[\leadsto \color{blue}{\sqrt[3]{\left(2 \cdot a\right) \cdot g}} \]
7. Add Preprocessing

Alternative 11: 3.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\left(a + a\right) \cdot g\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (pow (* (+ a a) g) 0.3333333333333333))

C

double code(double g, double a) {
	return pow(((a + a) * g), 0.3333333333333333);
}

Fortran

real(8) function code(g, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: a
    code = ((a + a) * g) ** 0.3333333333333333d0
end function

Java

public static double code(double g, double a) {
	return Math.pow(((a + a) * g), 0.3333333333333333);
}

Python

def code(g, a):
	return math.pow(((a + a) * g), 0.3333333333333333)

Julia

function code(g, a)
	return Float64(Float64(a + a) * g) ^ 0.3333333333333333
end

MATLAB

function tmp = code(g, a)
	tmp = ((a + a) * g) ^ 0.3333333333333333;
end

Wolfram

code[g_, a_] := N[Power[N[(N[(a + a), $MachinePrecision] * g), $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 69.9%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. pow1/3 N/A

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

   3. lower-pow.f64 33.2

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

4. Applied rewrites 1.2%

   \[\leadsto \color{blue}{{\left(\left(-2 \cdot a\right) \cdot g\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. unpow-1 N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow-prod-down N/A

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

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

   12. inv-pow N/A

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

   13. clear-num-rev N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg N/A

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

   16. associate-/l* N/A

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

   17. *-commutative N/A

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

   18. associate-/l* N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 3.2

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

6. Applied rewrites 3.2%

   \[\leadsto {\left(\color{blue}{\left(a + a\right)} \cdot g\right)}^{0.3333333333333333} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))