2-ancestry mixing, zero discriminant

Percentage Accurate: 75.4% → 98.7%

Time: 4.5s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

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

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

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

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

   5. neg-sub0 N/A

      \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot \color{blue}{\left(0 - g\right)}}}{\sqrt[3]{-a}} \]

   6. flip-- N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. neg-sub0 N/A

      \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(g \cdot g\right)}}{0 + g}}}{\sqrt[3]{-a}} \]

   10. lift-neg.f64 N/A

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

   11. +-lft-identity N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lift-neg.f64 N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. lift-neg.f64 N/A

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

   18. associate-/r* N/A

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

   19. *-inverses N/A

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

   20. lower-/.f64 N/A

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

   21. lift-neg.f64 N/A

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

   22. neg-mul-1 N/A

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

   23. associate-/r* N/A

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

   24. metadata-eval N/A

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

   25. lower-/.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. frac-2neg N/A

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

   14. cbrt-div N/A

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

   15. pow1/3 N/A

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

   16. pow1/3 N/A

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

   17. lower-/.f64 N/A

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

   18. pow1/3 N/A

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

   19. lower-cbrt.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. pow1/3 N/A

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

   24. lower-cbrt.f64 98.8

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

8. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 98.8

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

10. Applied rewrites 98.8%

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

Alternative 2: 83.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-307)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (cbrt (* (/ 0.5 a) g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-307) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = cbrt(((0.5 / a) * g));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-307) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.cbrt(((0.5 / a) * g));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-307)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = cbrt(Float64(Float64(0.5 / a) * g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-307], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(0.5 / a), $MachinePrecision] * g), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -5.00000000000000014e-307

   1. Initial program 78.8%

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

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

      7. cbrt-div N/A

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

      8. pow1/3 N/A

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

      9. associate-/l/ N/A

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

      10. neg-mul-1 N/A

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

      11. cbrt-prod N/A

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

      12. pow1/3 N/A

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

      13. times-frac N/A

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

      14. pow1/3 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. pow1/3 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval 92.1

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

   6. Applied rewrites 92.1%

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

   if -5.00000000000000014e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.1%

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

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. associate-/r/ N/A

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

      18. metadata-eval N/A

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

      19. lift-neg.f64 N/A

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

      20. frac-2neg N/A

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

      21. lift-/.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-cbrt.f64 74.1

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

      24. lift-*.f64 N/A

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

      25. lift-/.f64 N/A

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

   6. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot g}} \]
3. Recombined 2 regimes into one program.
4. 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 76.3%

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

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. cbrt-undiv N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}}} \]
7. 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 76.3%

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

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

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

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

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

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

   5. neg-sub0 N/A

      \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot \color{blue}{\left(0 - g\right)}}}{\sqrt[3]{-a}} \]

   6. flip-- N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. neg-sub0 N/A

      \[\leadsto \frac{\sqrt[3]{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(g \cdot g\right)}}{0 + g}}}{\sqrt[3]{-a}} \]

   10. lift-neg.f64 N/A

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

   11. +-lft-identity N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lift-neg.f64 N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. lift-neg.f64 N/A

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

   18. associate-/r* N/A

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

   19. *-inverses N/A

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

   20. lower-/.f64 N/A

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

   21. lift-neg.f64 N/A

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

   22. neg-mul-1 N/A

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

   23. associate-/r* N/A

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

   24. metadata-eval N/A

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

   25. lower-/.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. metadata-eval N/A

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

   10. associate-/r* N/A

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

   11. lift-neg.f64 N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. clear-num N/A

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

   15. cbrt-div N/A

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

   16. pow1/3 N/A

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

   17. lower-/.f64 N/A

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

   18. pow1/3 N/A

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

   19. lower-cbrt.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. div-inv N/A

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

   22. metadata-eval N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 98.7

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

8. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]
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 76.3%

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

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lift-/.f64 N/A

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

   14. associate-/r/ N/A

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

   15. cbrt-prod N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cbrt.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-cbrt.f64 98.6

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

6. Applied rewrites 98.6%

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

Alternative 6: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. lift-/.f64 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-neg.f64 N/A

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

   17. associate-/r/ N/A

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

   18. metadata-eval N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. lift-cbrt.f64 76.3

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

   24. lift-*.f64 N/A

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

   25. lift-/.f64 N/A

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

6. Applied rewrites 76.3%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))