2-ancestry mixing, zero discriminant

Percentage Accurate: 75.6% → 98.7%

Time: 6.2s

Alternatives: 6

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 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, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. 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 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{-0.5 \cdot g}}{\sqrt[3]{-a}}} \]
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-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

   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. associate-*r/ N/A

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

   11. associate-*l/ N/A

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

   12. cbrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. pow1/3 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/3 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. distribute-neg-frac2 N/A

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

   15. remove-double-neg N/A

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

   16. lower-/.f64 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{g \cdot -0.5} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -1e-303)
   (* (cbrt (* g -0.5)) (pow (- a) -0.3333333333333333))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-303) {
		tmp = cbrt((g * -0.5)) * pow(-a, -0.3333333333333333);
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-303) {
		tmp = Math.cbrt((g * -0.5)) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -1e-303)
		tmp = Float64(cbrt(Float64(g * -0.5)) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -1e-303], N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -9.99999999999999931e-304

   1. Initial program 72.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 91.8

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

   6. Applied rewrites 91.8%

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

   if -9.99999999999999931e-304 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 81.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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 98.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-0.5 \cdot g}}{\sqrt[3]{-a}}} \]
   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-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. cbrt-prod N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-cbrt.f64 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.0

          \[\leadsto {\left(2 \cdot a\right)}^{\color{blue}{-0.3333333333333333}} \cdot \sqrt[3]{g} \]

   8. Applied rewrites 92.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{g \cdot -0.5} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{-301}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 1e-301)
   (/ 1.0 (cbrt (* (/ a g) 2.0)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 1e-301) {
		tmp = 1.0 / cbrt(((a / g) * 2.0));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 1e-301) {
		tmp = 1.0 / Math.cbrt(((a / g) * 2.0));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 1e-301)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a / g) * 2.0)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 1e-301], N[(1.0 / N[Power[N[(N[(a / g), $MachinePrecision] * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 1.00000000000000007e-301

   1. Initial program 72.6%

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

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

      14. lift-cbrt.f64 N/A

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

      15. lift-cbrt.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cbrt.f64 N/A

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

      19. lift-cbrt.f64 N/A

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

      20. cbrt-undiv N/A

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

      21. lift-/.f64 N/A

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

      22. clear-num N/A

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

      23. associate-/l/ N/A

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

      24. metadata-eval N/A

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

      25. times-frac N/A

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

      26. lift-*.f64 N/A

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

      27. neg-mul-1 N/A

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

      28. lift-neg.f64 N/A

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

   6. Applied rewrites 73.2%

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

   if 1.00000000000000007e-301 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 81.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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 98.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-0.5 \cdot g}}{\sqrt[3]{-a}}} \]
   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-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. cbrt-prod N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-cbrt.f64 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.1

          \[\leadsto {\left(2 \cdot a\right)}^{\color{blue}{-0.3333333333333333}} \cdot \sqrt[3]{g} \]

   8. Applied rewrites 92.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{-301}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 77.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. 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 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{-0.5 \cdot g}}{\sqrt[3]{-a}}} \]
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-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

   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. associate-*r/ N/A

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

   11. associate-*l/ N/A

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

   12. cbrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 5: 75.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. 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 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{-0.5 \cdot g}}{\sqrt[3]{-a}}} \]
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. cbrt-undiv N/A

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

   14. lift-cbrt.f64 N/A

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

   15. lift-cbrt.f64 N/A

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

   16. clear-num N/A

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

   17. lower-/.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. lift-cbrt.f64 N/A

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

   20. cbrt-undiv N/A

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

   21. lift-/.f64 N/A

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

   22. clear-num N/A

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

   23. associate-/l/ N/A

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

   24. metadata-eval N/A

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

   25. times-frac N/A

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

   26. lift-*.f64 N/A

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

   27. neg-mul-1 N/A

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

   28. lift-neg.f64 N/A

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

6. Applied rewrites 77.7%

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

7. Final simplification 77.7%

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

Alternative 6: 75.6% 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 77.0%

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

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

   4. associate-/r/ N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow2 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 77.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 77.0

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

6. Applied rewrites 77.0%

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

Reproduce

herbie shell --seed 2024244 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))