2-ancestry mixing, zero discriminant

Percentage Accurate: 76.5% → 98.7%

Time: 3.2s

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: 76.5% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

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

   5. lift-cbrt.f64 N/A

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

   6. cbrt-div N/A

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

   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. lift-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 98.6

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

   15. lift-/.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lift-cbrt.f64 N/A

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

   18. cbrt-undiv N/A

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

   19. lower-cbrt.f64 N/A

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

   20. lower-/.f64 98.7

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in g around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cbrt.f64 N/A

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

   3. lower-cbrt.f64 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 2: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt[3]{-0.5 \cdot g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-308)
   (* (cbrt (* -0.5 g)) (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) <= -5e-308) {
		tmp = cbrt((-0.5 * g)) * 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) <= -5e-308) {
		tmp = Math.cbrt((-0.5 * g)) * 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) <= -5e-308)
		tmp = Float64(cbrt(Float64(-0.5 * g)) * (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], -5e-308], N[(N[Power[N[(-0.5 * g), $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) < -4.99999999999999955e-308

   1. Initial program 73.3%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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 98.5%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-1}}{\sqrt[3]{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
   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. lift-neg.f64 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      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 92.0

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

   6. Applied rewrites 92.0%

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

   if -4.99999999999999955e-308 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.6%

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

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 91.9

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

   8. Applied rewrites 91.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt[3]{-0.5 \cdot g} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-308)
   (/ 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) <= -5e-308) {
		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) <= -5e-308) {
		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) <= -5e-308)
		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], -5e-308], 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) < -4.99999999999999955e-308

   1. Initial program 73.3%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 98.6%

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

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. cbrt-undiv N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. div-inv N/A

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

      19. lower-cbrt.f64 N/A

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

      20. div-inv N/A

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

      21. lift-/.f64 N/A

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

      22. clear-num N/A

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

      23. lift-/.f64 N/A

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

      24. frac-times N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 74.7%

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

   if -4.99999999999999955e-308 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.6%

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

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 91.9

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

   8. Applied rewrites 91.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

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 74.0%

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

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}} \]
8. 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 74.0%

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

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. *-commutative N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. associate-/r/ N/A

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

   17. unpow-prod-down N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 98.7%

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

Alternative 6: 76.5% 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 74.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 74.0

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

4. Applied rewrites 74.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. clear-num N/A

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

   20. metadata-eval N/A

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

   21. distribute-neg-frac N/A

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

   22. lift-/.f64 N/A

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

   23. distribute-frac-neg2 N/A

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

   24. metadata-eval N/A

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

   25. lift-/.f64 N/A

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

   26. frac-2neg N/A

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

   27. metadata-eval N/A

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

   28. remove-double-div N/A

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

6. Applied rewrites 74.1%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))