2-ancestry mixing, zero discriminant

Percentage Accurate: 76.6% → 98.7%

Time: 5.2s

Alternatives: 5

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: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

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 76.8%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. cbrt-div N/A

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

   5. neg-mul-1 N/A

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

   6. cbrt-prod N/A

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

   7. pow1/3 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/3 N/A

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

   11. sqr-pow N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow-prod-down N/A

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

   17. sqr-pow N/A

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

   18. pow1/3 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. pow1/3 N/A

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

   22. lower-cbrt.f64 N/A

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

   23. lower-cbrt.f64 1.6

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

   24. remove-double-div N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-prod N/A

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

   9. neg-mul-1 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 2: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{-1}{\frac{-1}{g}}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 5e-307)
   (cbrt (/ (/ -1.0 (/ -1.0 g)) (* 2.0 a)))
   (* (cbrt (* 0.5 g)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-307) {
		tmp = cbrt(((-1.0 / (-1.0 / g)) / (2.0 * a)));
	} else {
		tmp = cbrt((0.5 * g)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-307) {
		tmp = Math.cbrt(((-1.0 / (-1.0 / g)) / (2.0 * a)));
	} else {
		tmp = Math.cbrt((0.5 * g)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 5e-307)
		tmp = cbrt(Float64(Float64(-1.0 / Float64(-1.0 / g)) / Float64(2.0 * a)));
	else
		tmp = Float64(cbrt(Float64(0.5 * g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 5e-307], N[Power[N[(N[(-1.0 / N[(-1.0 / g), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(0.5 * g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 5.00000000000000014e-307

   1. Initial program 76.6%

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

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

      4. *-commutative 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 76.5%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.5}{a} \cdot \left(-g\right)}} \]
   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. *-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. *-inverses N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. times-frac N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. div-inv N/A

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

      15. rgt-mult-inverse N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 76.6

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

   8. Applied rewrites 76.6%

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

   if 5.00000000000000014e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 77.1%

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

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

      4. cbrt-div N/A

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

      5. neg-mul-1 N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/3 N/A

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

      11. sqr-pow N/A

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

      12. pow-prod-down N/A

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

      13. sqr-neg N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. pow-prod-down N/A

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

      17. sqr-pow N/A

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

      18. pow1/3 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-cbrt.f64 N/A

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

      21. pow1/3 N/A

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

      22. lower-cbrt.f64 N/A

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

      23. lower-cbrt.f64 1.7

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

      24. remove-double-div N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. cbrt-undiv N/A

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

      6. lift-cbrt.f64 N/A

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

      7. cbrt-undiv N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. /-rgt-identity N/A

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

      12. un-div-inv N/A

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

      13. cbrt-prod N/A

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

      14. pow1/3 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. metadata-eval N/A

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

      18. lift-/.f64 N/A

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

      19. pow1/3 N/A

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

      20. *-commutative N/A

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

      21. unpow-prod-down N/A

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

   6. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 3: 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 76.8%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. cbrt-div N/A

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

   5. neg-mul-1 N/A

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

   6. cbrt-prod N/A

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

   7. pow1/3 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/3 N/A

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

   11. sqr-pow N/A

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

   12. pow-prod-down N/A

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

   13. sqr-neg N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. pow-prod-down N/A

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

   17. sqr-pow N/A

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

   18. pow1/3 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. pow1/3 N/A

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

   22. lower-cbrt.f64 N/A

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

   23. lower-cbrt.f64 1.6

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

   24. remove-double-div N/A

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-undiv N/A

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

   6. lift-cbrt.f64 N/A

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

   7. cbrt-undiv N/A

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

   8. frac-2neg N/A

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

   9. lift-neg.f64 N/A

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

   10. metadata-eval N/A

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

   11. /-rgt-identity N/A

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

   12. un-div-inv N/A

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

   13. *-commutative N/A

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

   14. cbrt-prod N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. lift-/.f64 N/A

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

   19. cbrt-prod N/A

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

   20. lift-neg.f64 N/A

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

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

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

6. Applied rewrites 98.7%

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

Alternative 4: 76.6% 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 76.8%

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

Alternative 5: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

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

   4. *-commutative 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 76.8%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{-0.5}{a} \cdot \left(-g\right)}} \]
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. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. neg-mul-1 N/A

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

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

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

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

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

   12. lift-neg.f64 N/A

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

   13. remove-double-neg N/A

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

   14. lower-*.f64 N/A

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

   15. distribute-frac-neg2 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-/r* N/A

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

   18. metadata-eval N/A

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

   19. distribute-neg-frac N/A

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

   20. lower-/.f64 N/A

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

   21. metadata-eval 76.8

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

6. Applied rewrites 76.8%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))