2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 5.8s

Alternatives: 12

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. lift-neg.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. cbrt-undiv N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. un-div-inv N/A

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

   11. clear-num N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. *-inverses N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. *-inverses N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 98.8

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

8. Applied rewrites 98.8%

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

Alternative 2: 91.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -2e-303)
   (* (cbrt (* -0.5 g)) (pow (- a) -0.3333333333333333))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-303) {
		tmp = cbrt((-0.5 * g)) * pow(-a, -0.3333333333333333);
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-303) {
		tmp = Math.cbrt((-0.5 * g)) * Math.pow(-a, -0.3333333333333333);
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -2e-303)
		tmp = Float64(cbrt(Float64(-0.5 * g)) * (Float64(-a) ^ -0.3333333333333333));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -2e-303], N[(N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.99999999999999986e-303

   1. Initial program 69.4%

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

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

      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 92.1

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

   6. Applied rewrites 92.1%

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

   if -1.99999999999999986e-303 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 73.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. cbrt-prod N/A

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

      14. cbrt-unprod N/A

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

      15. lift-cbrt.f64 N/A

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

      16. pow1/3 N/A

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

      17. 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 98.7%

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

      1. lift-cbrt.f64 N/A

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

      2. 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. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-/r/ 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

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

      12. lift-/.f64 N/A

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

      13. pow1/3 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      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. lower-*.f64 N/A

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

      21. metadata-eval 92.1

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

   8. Applied rewrites 92.1%

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

4. Final simplification 92.1%

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

Alternative 3: 84.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -2e-303)
   (cbrt (/ (/ 1.0 (/ (/ -1.0 g) -0.5)) a))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-303) {
		tmp = cbrt(((1.0 / ((-1.0 / g) / -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) <= -2e-303) {
		tmp = Math.cbrt(((1.0 / ((-1.0 / g) / -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) <= -2e-303)
		tmp = cbrt(Float64(Float64(1.0 / Float64(Float64(-1.0 / g) / -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], -2e-303], N[Power[N[(N[(1.0 / N[(N[(-1.0 / g), $MachinePrecision] / -0.5), $MachinePrecision]), $MachinePrecision] / a), $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) < -1.99999999999999986e-303

   1. Initial program 69.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. sqr-neg N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. pow2 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. *-inverses N/A

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

      9. times-frac N/A

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

      10. associate-*r* N/A

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

      11. lift-neg.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. clear-num N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. associate-*l/ N/A

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

      19. associate-/r* N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 69.4%

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

   if -1.99999999999999986e-303 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 73.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. cbrt-div N/A

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

      7. distribute-neg-frac N/A

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

      8. cbrt-div N/A

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

      9. pow1/3 N/A

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

      10. sqr-pow N/A

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

      11. pow-prod-down N/A

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

      12. sqr-neg N/A

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

      13. remove-double-neg N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/3 N/A

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

      18. cbrt-div N/A

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

      19. pow1/3 N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. cbrt-prod N/A

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

      14. cbrt-unprod N/A

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

      15. lift-cbrt.f64 N/A

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

      16. pow1/3 N/A

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

      17. 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 98.7%

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

      1. lift-cbrt.f64 N/A

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

      2. 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. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-/r/ 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

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

      12. lift-/.f64 N/A

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

      13. pow1/3 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      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. lower-*.f64 N/A

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

      21. 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 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. lift-neg.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. cbrt-undiv N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. un-div-inv N/A

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

   11. clear-num N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 5: 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 71.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. lift-neg.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. cbrt-undiv N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. un-div-inv N/A

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

   11. clear-num N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. *-inverses N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. *-inverses N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 98.8

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

8. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. remove-double-neg N/A

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

   5. neg-mul-1 N/A

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

   6. cbrt-div N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow-1 N/A

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

   12. remove-double-div N/A

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

   13. frac-times N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/r/ N/A

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

   18. lift-/.f64 N/A

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

   19. cbrt-div N/A

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

   20. lift-cbrt.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. lower-cbrt.f64 98.7

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

10. Applied rewrites 98.7%

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

Alternative 6: 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 71.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. cbrt-prod N/A

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

   14. cbrt-unprod N/A

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

   15. lift-cbrt.f64 N/A

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

   16. pow1/3 N/A

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

   17. 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 98.7%

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

Alternative 7: 76.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow2 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 71.4%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. *-inverses N/A

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

   9. times-frac N/A

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

   10. associate-*r* N/A

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

   11. lift-neg.f64 N/A

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

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

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

   13. lift-*.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. clear-num N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. associate-*l/ N/A

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

   19. associate-/r* N/A

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

   20. lower-/.f64 N/A

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

6. Applied rewrites 71.5%

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

Alternative 8: 76.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow2 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

4. Applied rewrites 71.4%

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

   1. *-rgt-identity N/A

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

   2. *-inverses N/A

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

   3. associate-/l* N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. clear-num N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lift-neg.f64 N/A

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

   16. associate-/r* N/A

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

   17. *-inverses N/A

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

   18. metadata-eval N/A

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

   19. lift-neg.f64 N/A

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

   20. neg-mul-1 N/A

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

   21. associate-/r* N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-/.f64 71.5

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

6. Applied rewrites 71.5%

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

7. Final simplification 71.5%

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

Alternative 9: 76.4% 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 71.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 71.5%

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

Alternative 10: 76.4% 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 71.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 71.5%

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

Alternative 11: 76.4% 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]

Derivation

1. Initial program 71.5%

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

Alternative 12: 76.4% 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 71.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. cbrt-div N/A

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

   7. distribute-neg-frac N/A

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

   8. cbrt-div N/A

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

   9. pow1/3 N/A

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

   10. sqr-pow N/A

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

   11. pow-prod-down N/A

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

   12. sqr-neg N/A

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

   13. remove-double-neg N/A

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

   14. remove-double-neg N/A

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

   15. pow-prod-down N/A

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

   16. sqr-pow N/A

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

   17. pow1/3 N/A

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

   18. cbrt-div N/A

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

   19. pow1/3 N/A

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

   20. lower-/.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lift-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-cbrt.f64 71.4

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

   16. lift-*.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. lift-/.f64 N/A

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

   20. associate-*l/ N/A

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

   21. lift-*.f64 N/A

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

   22. distribute-frac-neg2 N/A

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lift-neg.f64 N/A

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

6. Applied rewrites 71.4%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))