2-ancestry mixing, zero discriminant

Percentage Accurate: 76.2% → 98.7%

Time: 5.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

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 75.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 1.6

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

   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-unprod 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. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 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: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -2 \cdot 10^{-307}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -2e-307)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-307) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-307) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -2e-307)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -2e-307], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.99999999999999982e-307

   1. Initial program 75.4%

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

   4. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-neg.f64 91.8

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

   6. Applied rewrites 91.8%

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

   if -1.99999999999999982e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.7%

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 92.4%

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

Alternative 3: 83.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -2e-307)
   (cbrt (/ g (/ -1.0 (/ -0.5 a))))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-307) {
		tmp = cbrt((g / (-1.0 / (-0.5 / a))));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -2e-307) {
		tmp = Math.cbrt((g / (-1.0 / (-0.5 / a))));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -2e-307)
		tmp = cbrt(Float64(g / Float64(-1.0 / Float64(-0.5 / a))));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -2e-307], N[Power[N[(g / N[(-1.0 / N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.99999999999999982e-307

   1. Initial program 75.4%

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

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

      17. pow-sqr 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

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

   4. Applied rewrites 75.5%

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

   if -1.99999999999999982e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 74.7%

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 92.4%

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

Alternative 4: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{2 \cdot \frac{a}{g}}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 1.6

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

   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%

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

6. Applied rewrites 75.8%

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

7. Final simplification 75.8%

   \[\leadsto {\left(\sqrt[3]{2 \cdot \frac{a}{g}}\right)}^{-1} \]
8. Add Preprocessing

Alternative 5: 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 75.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 1.6

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

   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. div-inv N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. cbrt-div N/A

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

   12. metadata-eval N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

   15. pow1/3 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-cbrt.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. cbrt-undiv N/A

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

   20. pow1/3 N/A

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

   21. clear-num N/A

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

   22. lift-/.f64 N/A

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

   23. unpow-prod-down N/A

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

6. Applied rewrites 98.7%

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

Alternative 6: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 1.6

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

   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. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-undiv N/A

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

   9. un-div-inv N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. clear-num N/A

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

   15. lift-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-cbrt.f64 N/A

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

   18. cbrt-div N/A

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

   19. associate-*l/ N/A

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

6. Applied rewrites 75.1%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))