2-ancestry mixing, zero discriminant

Percentage Accurate: 77.3% → 98.7%

Time: 7.1s

Alternatives: 9

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. remove-double-div N/A

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

   2. inv-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. pow2 N/A

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

   8. remove-double-neg N/A

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

   9. remove-double-neg N/A

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

   10. sqr-neg N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-up N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. inv-pow N/A

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

   19. clear-num N/A

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

   20. /-rgt-identity N/A

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

   21. neg-sub0 N/A

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

4. Applied rewrites 45.6%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}}} \]
5. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

   3. frac-2neg N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   6. lower-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   7. lower-neg.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   8. lower-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)} \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(4 \cdot \frac{\frac{1}{2}}{a}\right)}\right)}} \]

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

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right) \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right)} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}} \]

   14. associate-*l* N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   17. lift-/.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right)\right)\right)}} \]

   18. associate-*r/ N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot \frac{1}{2}}{a}}\right)\right)\right)}} \]

   19. metadata-eval N/A

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

   20. distribute-neg-frac N/A

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

   21. lower-/.f64 N/A

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

   22. metadata-eval 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. cbrt-prod N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-/.f64 N/A

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

   16. div-inv N/A

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

   17. associate-*l* N/A

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

   18. lft-mult-inverse N/A

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

   19. metadata-eval N/A

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

   20. lower-cbrt.f64 N/A

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

   21. metadata-eval N/A

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

   22. lft-mult-inverse N/A

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

   23. *-commutative N/A

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

8. Applied rewrites 98.8%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-306)
   (* (pow (* a -2.0) -0.3333333333333333) (cbrt (- g)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = pow((a * -2.0), -0.3333333333333333) * cbrt(-g);
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = Math.pow((a * -2.0), -0.3333333333333333) * Math.cbrt(-g);
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-306)
		tmp = Float64((Float64(a * -2.0) ^ -0.3333333333333333) * cbrt(Float64(-g)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-306], N[(N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.00000000000000011e-306

   1. Initial program 76.0%

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

      1. remove-double-div N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. remove-double-neg N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. clear-num N/A

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

      20. /-rgt-identity N/A

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

      21. neg-sub0 N/A

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

   4. Applied rewrites 41.8%

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

      3. frac-2neg N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

      4. cbrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

      6. lower-cbrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

      8. lower-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)} \cdot \frac{\frac{1}{2}}{a}\right)}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(4 \cdot \frac{\frac{1}{2}}{a}\right)}\right)}} \]

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

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right) \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right)} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}} \]

      14. associate-*l* N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right)\right)\right)}} \]

      18. associate-*r/ N/A

          \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot \frac{1}{2}}{a}}\right)\right)\right)}} \]

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 98.6

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

   8. Applied rewrites 92.0%

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

   if -4.00000000000000011e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 68.8%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.4

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

   4. Applied rewrites 92.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g \cdot -0.5} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-306)
   (* (cbrt (* g -0.5)) (pow (- a) -0.3333333333333333))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = cbrt((g * -0.5)) * pow(-a, -0.3333333333333333);
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = Math.cbrt((g * -0.5)) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-306)
		tmp = Float64(cbrt(Float64(g * -0.5)) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-306], N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.00000000000000011e-306

   1. Initial program 76.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. pow1/3 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-/r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. pow-prod-down N/A

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

      16. pow1/3 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cbrt.f64 N/A

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

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

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval N/A

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

   8. Applied rewrites 91.9%

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

   if -4.00000000000000011e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 68.8%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.4

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

   4. Applied rewrites 92.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g \cdot -0.5} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-306)
   (cbrt (* g (* (* (sqrt 0.5) (sqrt 0.5)) (/ 1.0 a))))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = cbrt((g * ((sqrt(0.5) * sqrt(0.5)) * (1.0 / a))));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-306) {
		tmp = Math.cbrt((g * ((Math.sqrt(0.5) * Math.sqrt(0.5)) * (1.0 / a))));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -4e-306)
		tmp = cbrt(Float64(g * Float64(Float64(sqrt(0.5) * sqrt(0.5)) * Float64(1.0 / a))));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -4e-306], N[Power[N[(g * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.00000000000000011e-306

   1. Initial program 76.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 76.0

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

   4. Applied rewrites 76.0%

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

      1. rem-square-sqrt N/A

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

      2. pow1/2 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. unpow-prod-down N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. unpow-prod-down N/A

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

      10. swap-sqr N/A

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

      11. pow1/2 N/A

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

      12. inv-pow N/A

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

      13. sqrt-pow1 N/A

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. sqrt-pow1 N/A

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

      17. sqr-pow N/A

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

      18. inv-pow N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. pow1/2 N/A

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

      22. lower-sqrt.f64 N/A

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

      23. pow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. lower-/.f64 76.0

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

   6. Applied rewrites 76.0%

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

   if -4.00000000000000011e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 68.8%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.4

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

   4. Applied rewrites 92.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. remove-double-div N/A

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

   2. inv-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. pow2 N/A

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

   8. remove-double-neg N/A

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

   9. remove-double-neg N/A

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

   10. sqr-neg N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-up N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. inv-pow N/A

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

   19. clear-num N/A

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

   20. /-rgt-identity N/A

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

   21. neg-sub0 N/A

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

4. Applied rewrites 45.6%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}}} \]
5. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}}} \]

   3. frac-2neg N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   6. lower-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   7. lower-neg.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{neg}\left(g\right)}}}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   8. lower-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}\right)}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)} \cdot \frac{\frac{1}{2}}{a}\right)}} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(4 \cdot \frac{\frac{1}{2}}{a}\right)}\right)}} \]

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

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right) \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{\left(a \cdot a\right)} \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)}} \]

   14. associate-*l* N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{\color{blue}{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(4 \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)}}} \]

   17. lift-/.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(4 \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right)\right)\right)}} \]

   18. associate-*r/ N/A

       \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\sqrt[3]{a \cdot \left(a \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot \frac{1}{2}}{a}}\right)\right)\right)}} \]

   19. metadata-eval N/A

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

   20. distribute-neg-frac N/A

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

   21. lower-/.f64 N/A

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

   22. metadata-eval 98.7

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

6. Applied rewrites 98.7%

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 7: 77.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 72.3

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

4. Applied rewrites 72.3%

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

   1. rem-square-sqrt N/A

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

   2. pow1/2 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. unpow-prod-down N/A

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

   6. pow1/2 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. unpow-prod-down N/A

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

   10. swap-sqr N/A

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

   11. pow1/2 N/A

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

   12. inv-pow N/A

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

   13. sqrt-pow1 N/A

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

   14. pow1/2 N/A

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

   15. inv-pow N/A

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

   16. sqrt-pow1 N/A

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

   17. sqr-pow N/A

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

   18. inv-pow N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. pow1/2 N/A

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

   22. lower-sqrt.f64 N/A

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

   23. pow1/2 N/A

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

   24. lower-sqrt.f64 N/A

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

   25. lower-/.f64 72.3

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

6. Applied rewrites 72.3%

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

7. Final simplification 72.3%

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

Alternative 8: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Final simplification 72.3%

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

Alternative 9: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 72.3

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

4. Applied rewrites 72.3%

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

5. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))