2-ancestry mixing, zero discriminant

Percentage Accurate: 76.8% → 98.6%

Time: 6.0s

Alternatives: 10

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: 76.8% 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: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 98.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. unpow-1 N/A

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

   6. lift-pow.f64 N/A

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

   7. clear-num N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 92.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-308)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-308) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-308) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-308)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-308], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

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

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \sqrt[3]{-0.5 \cdot g}} \]
   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. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 91.9

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

   6. Applied rewrites 91.9%

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

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

   1. Initial program 77.7%

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

   4. Applied rewrites 98.7%

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

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.5%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-1 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 92.1

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

   8. Applied rewrites 92.1%

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

Alternative 3: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{-0.5}{a}}{\frac{-1}{g}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 2e-276)
   (cbrt (/ (/ -0.5 a) (/ -1.0 g)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 2e-276) {
		tmp = cbrt(((-0.5 / a) / (-1.0 / g)));
	} 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) <= 2e-276) {
		tmp = Math.cbrt(((-0.5 / a) / (-1.0 / g)));
	} 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) <= 2e-276)
		tmp = cbrt(Float64(Float64(-0.5 / a) / Float64(-1.0 / g)));
	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], 2e-276], N[Power[N[(N[(-0.5 / a), $MachinePrecision] / N[(-1.0 / g), $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) < 2e-276

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. inv-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. pow-prod-down N/A

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

      14. sqr-neg N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. pow2 N/A

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

      18. pow-pow N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. inv-pow N/A

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

   4. Applied rewrites 79.2%

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

   if 2e-276 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.3%

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

   4. Applied rewrites 98.7%

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

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.4%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-1 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 4: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 98.7%

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

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 98.6%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-cbrt.f64 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 98.7%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. unpow-1 N/A

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

   6. lift-pow.f64 N/A

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

   7. clear-num N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 98.7

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

8. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. un-div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-cbrt.f64 N/A

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

   13. cbrt-div N/A

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

   14. lift-/.f64 N/A

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

   15. clear-num N/A

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

   16. cbrt-div N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. associate-*r/ N/A

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

   21. frac-2neg N/A

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

   22. lift-/.f64 N/A

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

   23. cbrt-prod N/A

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

10. Applied rewrites 98.7%

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

Alternative 7: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 98.7%

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

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 98.6%

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

Alternative 8: 76.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. associate-/r* N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. pow-prod-up N/A

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

   13. pow-prod-down N/A

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

   14. sqr-neg N/A

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

   15. remove-double-neg N/A

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

   16. remove-double-neg N/A

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

   17. pow2 N/A

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

   18. pow-pow N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. inv-pow N/A

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

4. Applied rewrites 78.0%

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

Alternative 9: 76.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. 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 78.0%

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

Alternative 10: 76.8% 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 78.0%

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

4. Applied rewrites 98.7%

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

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

   9. unpow-1 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-neg.f64 N/A

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

   15. div-inv N/A

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

   16. associate-/r* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-cbrt.f64 77.0

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

   20. lift-/.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-pow.f64 N/A

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

   24. unpow-1 N/A

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

6. Applied rewrites 78.0%

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

Reproduce

herbie shell --seed 2024263 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))