2-ancestry mixing, zero discriminant

Percentage Accurate: 76.2% → 98.7%

Time: 4.4s

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: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-306)
   (* (pow (- a) -0.3333333333333333) (cbrt (* g -0.5)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-306) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((g * -0.5));
	} 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-306) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((g * -0.5));
	} 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-306)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(g * -0.5)));
	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-306], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -0.5), $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) < -4.99999999999999998e-306

   1. Initial program 77.1%

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

   4. Applied rewrites 98.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval 92.2

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

   6. Applied rewrites 92.2%

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

   if -4.99999999999999998e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 81.5%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. cbrt-div N/A

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

      6. pow1/3 N/A

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

      7. associate-*r/ N/A

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

      8. lift-cbrt.f64 N/A

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

      9. pow1/3 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. unpow-prod-down N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. pow1/3 N/A

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

      20. cbrt-div N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 98.9%

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

      1. remove-double-div N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-/.f64 N/A

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

      9. cbrt-div N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow-flip N/A

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

      14. lower-pow.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 91.6

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

   8. Applied rewrites 91.6%

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

4. Final simplification 91.9%

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

Alternative 3: 84.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-306)
   (cbrt (/ g (/ -1.0 (/ -0.5 a))))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-306) {
		tmp = cbrt((g / (-1.0 / (-0.5 / a))));
	} 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-306) {
		tmp = Math.cbrt((g / (-1.0 / (-0.5 / a))));
	} 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-306)
		tmp = cbrt(Float64(g / Float64(-1.0 / Float64(-0.5 / a))));
	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-306], N[Power[N[(g / N[(-1.0 / N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $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) < -4.99999999999999998e-306

   1. Initial program 77.1%

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

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg2 N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up 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-down N/A

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

      13. sqr-neg N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. pow2 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

      22. lower-/.f64 N/A

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

   4. Applied rewrites 77.1%

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

   if -4.99999999999999998e-306 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 81.5%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. cbrt-div N/A

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

      6. pow1/3 N/A

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

      7. associate-*r/ N/A

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

      8. lift-cbrt.f64 N/A

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

      9. pow1/3 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. unpow-prod-down N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. pow1/3 N/A

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

      20. cbrt-div N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 98.9%

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

      1. remove-double-div N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-/.f64 N/A

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

      9. cbrt-div N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow-flip N/A

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

      14. lower-pow.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 91.6

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

   8. Applied rewrites 91.6%

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

Alternative 4: 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 79.3%

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. cbrt-div N/A

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

   6. pow1/3 N/A

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

   7. associate-*r/ N/A

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

   8. lift-cbrt.f64 N/A

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

   9. pow1/3 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. unpow-prod-down N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. clear-num N/A

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

   17. lift-/.f64 N/A

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

   18. associate-*l/ N/A

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

   19. pow1/3 N/A

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

   20. cbrt-div N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 5: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.6

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

4. Applied rewrites 79.6%

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

Alternative 6: 76.2% 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 79.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.6

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

4. Applied rewrites 79.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. lift-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 79.6

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

6. Applied rewrites 79.6%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))